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INTRODUCTION 


ELEMENTS  OF  ALGEBRA. 


^=^-^S^ 


AN 


mtRODUCTION 


ELEMENTS  OF  ALGEBRA, 


DESIGNED  FOB  THE  USE  OF  THOSE 


I     WHO  ARE  ACQUAINTED  ONLY  WITH  THE  FIRST  PRINCIPLES 


ARITHMETIC. 


Selected  from  the  Algebra  of  Euler. 


CAMBRIDGE,  N.  ENG. 

PRINTED  BY  HILLIARD  AND  METCAIF, 

At  the  Unirersity  Press. 

SOI.D  BT  W.HILLIARD,  CAMBRIDGE,  AND   BT  CUMHIBTGS  AND  BILLIARD, 

NO.  1  CORNHILL,  BOSTON. 

1818. 


DISTRICT  OF  MASSACHUSETTS,  TO  WIT : 

District  Clerk's  Office. 

BE  it  remembered,  that  on  the  ninth  day  of  February  A.  D.  1818,  and  in  the  forty  second  year 
of  the  Independence  of  the  United  States  of  America,  JOHN  FARRAR  of  the  said  District  has  de- 
posited in  tnis  office  the  title  of  a  book,  the  right  whereof  he  claims  as  proprietor,  in  the  words  fol- 
lo^»^ng,  to  wit  : 

•  An  Introduction  to  the  Elements  of  Algebra,  designed  for  the  use  of  those  who  are  acquainted 
only  with  the  first  pi-incipies  of  Arithmetic.    Selected  from  the  Algebra  of  Euler.' 

In  conformity  to  the  act  of  the  Congress  of  the  United  States,  entitled,  "  An  Act  for  the  en- 
couragement of  learning,  dy  securing  the  copies  of  maps,  charts  and  books,  to  the  authors  and 
proprietors  of  such  copies,  during  the  times  therein  mentioned  :"  and  also  to  an  Act,  entitled 
"  An  Act  supplementary  to  an  Act,  entitled  An  Act  for  the  encouragement  of  learning,  by 
securing  the  copies  of  maps,  charts  and  books,  to  the  authors  and  proprietors  of  such  copies 
during  the  times  therein  mentioned  :  and  extending  the  benefits  thereof  to  the  arts  of  designmg, 
engraving  and  etching  historicai  and  other  prints." 

Txin    iir    T\^\Trc   S  ^^^^  of  the  District 
mo.  W.  DAVIS,  ^  oj-j^asiachusetts. 


A 


^x. 


ADVERTISEMENT. 

None  but  those  who  are  just  entering  upon 
the  study  of  Mathematics  need  to  be  informed 
of  the  high  character  of  Euler's  Algebra.  It 
has  been  allowed  to  hold  the  very  first  place 
among  elementary  works  upon  this  subject. 
The  autlsor  was  a  man  of  genius.  He  did  not,  like 
most  writers,  compile  from  others.  He  wrote 
from  his  own  reflections.  He  simplified  and  im- 
proved what  was  known,  and  added  much  that 
was  new.  He  is  particularly  distinguished  for  tlie 
clearness  and  comprehensiveness  of  his  views. 
He  seems  to  have  the  subject  of  which  he  treats 
present  to  his  mind  in  all  its  relations  and 
bearings  before  he  begins  to  write.  The  parts 
of  it  are  arranged  in  the  most  admirable  order. 
Each  step  is  introduced  by  the  preceding,  and 
leads  to  that  which  follows,  and  the  whole  taken 
together  constitutes  an  entire  and  connected 
piece,  like  a  highly  wrought  story. 

This  author  is  remarkable  also  for  his  illus- 
trations. He  teaches  by  instances.  He  presents 
one  example   after   another,  each  evident  by 


vi  Advertisement. 

itself,  and  each  throwing  some  new  light  npon 
the  subject,  till  the  reader  begins  to  anticipate 
for  himself  the  truth  to  be  inculcated. 

Some  opinion  may  be  formed  of  the  adapta- 
tion of  this  treatise  to  learners,  from  the  cir- 
cumstances under  which  it  was  composed.  It 
was  undertaken  after  the  author  became  blind, 
and  was  dictated  to  a  young  man  entirely  with- 
out education,  who  by  this  means  became  an 
expert  algebraist,  and  was  able  to  render  the 
author  important  services  as  an  amanuensis. 
It  was  written  originally  in  German.  It  has 
since  been  translated  into  Russian,  French,  and 
English,  with  notes  and  additions. 

The  entire  work,  consists  of  two  volumes 
octavo,  and  contains  many  things  intended  for 
the  professed  matlicmatician,  rather  than  the 
general  student.  It  was  thought  that  a  selec- 
tion of  such  parts  as  would  form  an  easy  intro- 
duction to  the  science  would  be  well  received, 
and  tend  to  promote  a  taste  for  analysis  among 
the  higher  class  of  students,  and  to  raise  the 
character  of  mathematical  learning. 

Notwithstanding  the  high  estimation  in  which 
this  work  has  been  held,  it  is  scarcely  to  be  met 
with  in  the  country,  and  is  very  little  known  in 
England.  On  the  continent  of  Europe  this 
author  is  the  constant  theme  of  eulogy.  His 
writings  have  the  character  of  classics.  They 
are  regarded  at  the  same  time  as    the  most 


Mvertisement.  vU 

profound  and  the  most  perspicuous,  and  as 
affording  the  finest  models  of  analysis.  They 
furnish  the  germs  of  the  most  approved  ele- 
mentary works  on  the  different  branches  of  this 
science.  The  constant  reply  of  one  of  the  first 
mathematicians*  of  France  to  those  who  con- 
sulted him  upon  the  best  method  of  studying 
mathematics  was,  "  study  Euler.^^  "  It  is  need- 
less," said  he,  '  to  accumulate  books  ;  true 
lovers  of  mathematics  will  always  read  Euler ; 
because  in  his  writings  every  thing  is  clear, 
distinct,  and  correct ;  because  they  swarm  with 
excellent  examples ;  and  because  it  is  always 
necessary  to  have  recourse  to  the  fountain 
head." 

The  selections  here  offered  are  from  the  first 
English  edition.  A  few  errors  have  been  cor- 
rected and  a  few  alterations  made  in  the 
phraseology.  In  the  original  no  questions  were 
left  to  be  performed  by  the  learner.  A  collec- 
tion was  made  by  the  English  translator  and 
subjoined  at  the  end  with  references  to  the 
sections  to  whicli  they  relate.  These  have  been 
mostly  retained,  and  some  new  ones  have  been 
added. 

Although  this  work  is  intended  particularly 
for  the  algebraical  student,  it  will  be  found  to 
contain  a  clear  and  full  explanation  of  the  fun- 
damental principles  of  arithmetic  ;  vulgar  frac- 

•  Lagrange. 


\iu  Mveriisement 

tions,  the  doctrine  of  roots  and  powers,  of  the 
different  kinds  of  proportion  and  progression, 
are  treated  in  a  manner,  that  can  hardly  fail  to 
interest  the  learner  and  make  him  acquainted 
T\  ith  the  reason  of  those  rules  which  he  has  so 
frequent  occasion  to  apply. 

A  more  extended  work  on  Algebra  formed 
after  the  same  model  is  now  in  the  press  and  will 
soon  be  published,  This  will  be  followed  by 
other  treatises  upon  the  diflPerent  branches  of 

pure  mathematics. 

JOHN  FARRAR, 

Professor  of  :Mathematics  and  Natural  Philosophy  in  the 
§  UniTersity  at  Cambridge. 

Cambridge^  February,  1818. 


CONTENTS, 


SECTION  I. 
OP  THE  DIFFERENT  METHODS  OF  CALCULATING  SIMPLE  QUANTITIES. 

CHAPTER  I. 

€f  Mathematics  in  general,  1 

CHAPTER  II. 
Explanation  of  the  Sign>s  +  Plus  and  —  Minus.  -3 

CHAPTER  III. 
Of  the  Multiplication  of  Simple  Quantities,  6 

CHAPTER  IV. 
Of  the  nature  of  wJiole  J^umhers  or  Integers^  with  respect  to 

their  Factors,  9 

CHAPTER  V. 
Of  the  Division  of  Simple  ^lantities,  1 1 

CHAPTER  VI. 
Of  the  properties  of  Integers  with  respect  to  their  Divisors,  14 

CHAPTER  VII. 
Of  Fractions  in  general,  \7 

CHAPTER  VIII. 
Of  the  properties  of  Fractions,  22 

CHAPTER  IX. 
Of  the  Addition  and  Subtraction  of  Fractions,  24 

CHAPTER  X. 
Of  the  Multiplication  and  Division  of  Fractions,  27 

CHAPTER  XI. 
Of  Square  Mmhers,  31 

CHAPTER  XII. 
Of  Square  Roots,  and  of  Irrational  JVambers  residting  from 
them*  33 


X  (jontents. 

CHAPTER  XIII. 

Of  Impossible  o?-  Imaginary  Quantities)  whicJi  arise  from  the 

same  source.  58 

CHAPTER  XIV. 
Of  Cubic  lumbers.  41 

CHAPTER  XV. 
Of  Cube  Roots,  and  of  Irrational  J\'^umbers  resulting  from  them,    42 

CHxVPTER  XVI. 
Of  Powers  in  general,  4  5 

CHAPTER  XVII. 
Of  tJie  calculation  of  Powers,  49 

CHAPTER  XVIII. 
Of  Roots  with  rdation  to  Powers  in  general,  51 

CHAPTER  XIX. 
Of  the  Method  nf  representing  Irrational  JS*umbers  by  Fractional 
Exponents,  5S 

CHAPTER  XX. 
Cjf  the  different  Methods  of  CalculatioUf  and  of  their  mutual 

Connexion,  56 

SECTION  SECOND. 
OF  THE  DIFFERENT  METHODS  OF  CALCULATING  COMPOUNp  QUANTITIES. 

CHAPTER  I. 

Of  the  Addition  of  Compound  ^naniities,  59 

CHAPTER  il. 
Of  the  Subtraction  of  Compound  Quantities,  61 

CHAPTER  III. 
Of  the  Multiplication  of  Compound  Quantities,  62 

CHAPTER  IV. 
Of  the  Division  of  Compound  Quantities,  68 

CHAPTER  V. 
Cff  the  Resolution  of  Fractions  into  hifinite  Series,  72 

CHAPTER  VI. 
Of  the  Squares  of  Compound  Quantities,  81 

CHAPTER  VII. 
Of  the  Extraction  of  Roots  applied  to  Compound  (Quantities.        84 


Contents,  xi 

CHAPTER  VIII. 

(jftlie  caleidation  of  Irrational  Quantities,  88 

CHAPTER  IX. 
Of  Cubes,  and  of  the  Extraction  of  Cube  Roots,  92 

CHAPTER  X. 
Of  the  higher  Powers  of  Compoimd  Quantities,  94 

CHAPTER  XL 
Of  the  Transposition  of  the  Letters,  an  whicJf,  the  demonstra' 

tration  of  the  preceding  Rule  is  founded.  10# 

CHAPTER  XII. 
Of  the  expression  of  Irrational  Powers  by  Infinite  Series,  104 

CHAPTER  XIII. 
Of  tJie  resolution  of  Mgative  Powers,  107 

SECTION  THIRD. 

OF  RATIOS  AND  PROPORTIONS. 

CHAPTER  I. 

Of  Arithmetical  Ratio,  or  of  the  difference  between  two  JV^wm- 
bers.  111 

CHAPTER  II. 
Of  Arithmetical  Proportion,  1  is 

CHAPTER  III. 
Of  Arithmetical  Progressions,  1 16 

CHAPTER  IV. 
Of  the  Summation  of  Arithmetical  Progressions*  120 

CHAPTER  V. 
Of  Geometrical  Ratio,  124 

CHAPTER  VI. 
Of  the  greatest  Common  Divisor  of  two  given  numbers,  126 

CHAPTER  VII. 
Of  Geometrical  Proportions,  13© 

CHAPTER  VIII. 
Observations  on  the  Rules  of  Proportion  and  their  utilittj,  133 

CHAPTER  IX. 
Of  Compound  Relations,  '  13S 


xii  (Contents. 

CHAPTER  X. 

Of  Geometrical  Progressions.  144 

CHAPTER  XI. 

Of  Infinite  Decimal  Fractions.  150 

SFXTIOl^f  FOURTH. 
OF  ALGEBRAIC  EQUATIONS,  AND  OF  THE  RESOLUTION  OF  THOSE  EQUATIONS. 

CHAPTER  I. 

Of  the  Solution  of  Problems  in  general.  15S 

CHAPTER  II. 

Of  the  Resolution  of  Simple  Mquatiatis,  or  Equations  of  the 
first  degree,  159 

CHAPTER  III, 

Of  the  Solution  of  Questions  relating  to  the  preceding  chapter.       163 

CHAPTER  IV. 

Of  the  Resolution  of  two  or  more  Equations  of  the  First  Degree,    1 73 

CHAPTER  V. 
Of  the  Resolutim  (f  Pure  Quadratic  Equations.  182 

CHAPTER  VI. 
Of  the  Resolution  ofMixt  Equations  of  the  Second  Degree.         18S 

CHAPTER  VII. 
Of  the  JVature  of  Equations  of  the  Second  Degree.  196 

Questions  for  Practice^  202 

A'^otes.  -214 


INTRODUCTION 


ELEMENTS  OF  ALGEBRA. 


SECTION  I. 
OF  THE  DIFFERENT  METHODS  OF  CALCULATING  SIMPLE  QUANTITIES. 

CHAPTER  I. 

OJ  Mathematics  in  generaU 

ARTICIE    I. 

Whatever  is  capable  of  increase,  or  diminution,  is  called 
magnitude  or  quantity. 

A  sum  of  money  therefore  is  a  quantity,  since  we  may  in- 
crease it  and  diminish  it.  It  is  the  same  with  a  weight,  and 
other  things  of  this  nature. 

2.  From  this  definition,  it  is  evident,  that  the  different  kinds 
of  magnitude  must  be  so  many  as  to  render  it  difficult  even  to 
enumerate  them  :  and  this  is  tlie  origin  of  tlie  different  branches 
of  the  Mathematics,  each  being  employed  on  a  particular  kind 
of  magnitude.  Mathematics,  in  general,  is  the  science  of  quan- 
tity ;  or  the  science  whicli  investigates  the  means  of  measuring 
quantity. 

3.  Now  we  cannot  measure  or  determine  any  quantity, 
except  by  considering  some  other  quantity  of  the  same  kind 
as  known,  and  pointing  out  their  mutual  relation.  If  it  were 
proposed,  for  example,  to  determine  the  quantity  of  a  sum  of 
money,  we  should  take  some  known  piece  of  money  (as  a  louis, 
a  crown,  a  ducat,  or  some  other  coin,)  and  shew  how  many  of 

I 


g  Algebra.  Sect.  1. 

these  pieces  are  contained  in  the  given  sum.  In  the  same  man- 
ner, if  it  were  proposed  to  determine  the  quantity  of  a  weight, 
w^e  should  take  a  certain  known  weight ;  for  example,  a  pound, 
an  ounce,  &c.  and  then  shew  how  many  times  one  of  these 
weights  is  contained  in  that  which  we  are  endeavouring  to 
ascertain.  If  we  wished  to  measure  any  length  or  extension, 
•we  should  make  use  of  some  known  length,  such  as  a  foot. 

4.  So  that  the  determination,  or  the  measure  of  magnitude  of 
all  kinds,  is  reduced  to  this  :  fix  at  pleasure  upon  any  one  known 
magnitude  of  the  same  species  with  that  which  is  to  be  deter- 
mined, and  consider  it  as  the  measure  or  unit ;  then,  determine 
the  proportion  of  the  proposed  magnitude  to  this  known  mea- 
sure. This  proportion  is  always  expressed  by  numbers;  so 
that  a  number  is  nothing  but  the  proportion  of  one  magnitude 
to  another  arbitrarily  assumed  as  the  unit. 

5.  From  this  it  appears,  that  all  magnitudes  may  be  expressed 
by  numbers ;  and  that  the  foundation  of  all  the  mathematical 
sciences  must  be  laid  in  a  complete  treatise  on  the  science  of 
numbers,  and  in  an  accurate  examination  of  the  different  pos- 
sible methods  of  calculation. 

This  fundamental  part  of  mathematics  is  called  Analysis,  or 
Algebra,  [l.p 

6.  In  Algebra  then  we  consider  only  numbers  which  repre- 
sent quantities,  without  regarding  the  different  kinds  of  quantity. 
These  are  the  subjects  of  other  branches  of  the  mathematics. 

7.  Arithmetic  treats  of  numbers  in  particular,  and  is  the 
science  of  numbers  properly  so  called ;  but  this  science  extends 
only  to  certain  methods  of  calculation  which  occur  in  common 
practice ;  Algebra,  on  the  contrary,  comprehends  in  general 
all  the  cases  which  can  exist  in  the  doctrine,  and  calculation  of 
numbers. 

*  The  numbers  thus  included  in  crotchets  refer  to  notes  at  the 
end  of  this  introduction. 


Chap.  2.  Of  Simple  Quantities.  ^ 

CHAPTER  II. 

Explanation  of  the  Signs  +  Plus  and  —  Minus. 

8.  When  we  have  to  add  one  given  number  to  another,  this 
is  indicated  by  the  sign  +  which  is  placed  before  the  second 
number,  and  is  read  plus,  Tiius  5  -f  3  signifies  that  we  must 
add  3  to  the  number  5,  and  every  one  knows  that  the  result  is 
8  ;  in  the  same  manner  12  +  7  make  19  ;  25  -f  16  make  41 ;  the 
sum  of  25  +41  is  66,  &c. 

9.  We  also  make  use  of  the  same  sign  +  plus,  to  connect 
several  numbers  together ;  for  example,  r  +  5  +  9  signifies  that 
to  the  number  7  we  must  add  5  and  also  9,  which  make  21. 
The  reader  will  therefore  understand  what  is  meant  by 

8  +  5  +  13  +  11+1  +3  +  10; 
vi».  the  sum  of  all  these  numbers,  which  is  51. 

10.  All  this  is  evident ;  and  we  have  only  to  mention,  that, 
in  Algebra,  in  order  to  generalize  numbers,  we  represent  them 
by  letters,  as  a,  &,  c,  d,  &c.  Thus,  the  expression  a  +  6,  signifies 
the  sum  of  two  numbers,  which  we  express  by  a  and  6.  and 
these  numbers  may  be  eitlier  very  great  or  very  small.  In  the 
same  manner,  /  +  w  +  &  +  a?,  signifies  the  sum  of  the  numbers 
represented  by  these  four  letters. 

If  we  know  therefore  the  numbers  that  are  represented  by 
letters,  we  shall  at  all  times  be  able  to  find  by  arithmetic,  the 
sum  or  value  of  similar  expressions. 

11.  When  it  is  required,  on  the  contrary,  to  subtract  one 
given  number  from  another,  this  operation  is  denoted  by  the 
sign  — ,  wliich  signifies  miiius,  and  is  placed  before  the  number 
to  be  subtracted  :  thus  8  —  5  signifies  that  the  number  5  is  to  be 
taken  from  the  number  8 ;  which  being  done,  there  remains  3. 
In  like  manner  12  —  7  is  the  same  as  5  ;  and  20  — 14  is  the  same 
as  6,  &c. 

12.  Sometimes  also  we  may  have  several  numbers  to  subtract 
from  a  single  one  ;  as,  for  instance,  50  —  1  —  3 —  5—7—9.  This 
signifies,  first,  take  1  from  50,  there  remains  49  ;  take  3  from 
that  remainder,  there  will  remain  46  ;  take  away  5, 41  remains ; 
take  away  7,  34  remains ;  lastly,  from  that  take  9,  and  there 


4  Mgebra.  Sect.  I. 

remains  25 ;  this  last  remainder  is  the  value  of  the  expression. 
But  as  the  numbers  1,  3,  5,  7,  9,  are  all  to  be  subtracted,  it  is 
the  same  thing  if  we  subtract  their  sum,  which  is  25,  at  once 
from  50,  and  the  remainder  will  be  25  as  before. 

IS.  It  is  also  very  easy  to  determine  the  value  of  similar 
expressions,  in  which  both  the  signs  +  phis  and  —  minus  are 
found  :  for  example  ; 

12  —  3  —  5  -f-  2  —  1  is  the  same  as  5. 
We  have  only  to  collect  separately  the  sum  of  the  numbers  that 
have  the  sign  -\-  before  them,  and  subtract  from  it  the  sum  of 
those  that  have  the  sign  — .     The  sum  of  12  and  2  is  14  ,•  that 
of  3,  5  and  1,  is  9 ;  now  9  being  taken  from  14,  there  remains  5. 

14.  It  will  be  perceived  from  these  examples  that  the  order 
in  which  we  write  the  numhers  is  quite  indifferent  and  arbitrary^ 
provided  the  proper  sign  of  each  be  preserved,  AYe  might  with 
equal  propriety  have  arranged  the  expression  in  the  preceding 
article  thus  ;  12-f2  —  5  —  3  —  l,or2  —  1  —  3  —  5-fl2,  or2  + 
12  —  3  —  1  —  5,  or  in  still  different  orders.  It  must  be  observed, 
that  in  the  expression  proposed,  the  sign  -|-  is  supposed  to  be 
placed  before  the  number  12. 

15.  It  will  not  be  attended  with  any  more  difficulty,  if,  in 
order  to  generalize  these  operations,  we  make  use  of  letters 
instead  of  real  numbers.     It  is  evident,  for  example,  that 

(J  —  6  —  c+d  —  e, 
signifies  that  we  have  numbers  expressed  by  a  and  <Z,  and  that 
from  these  numbers,  or  from  their  sum,  we  must  subtract  the 
numbers  expressed  by  the  letters  &,  c,  e,  which  have  before  them 
the  sign  — . 

16.  Hence  it  is  absolutely  necessary  to  consider  what  sign  is 
prefixed  to  each  number :  for  in  algebra,  simple  quantities  are 
numbers  considered  with  regard  to  the  signs  which  precede,  or 
affect  them.  Further,  we  call  those  positive  quantities,  before 
which  the  sign  -f  is  found  ;  and  those  are  called  negative  quan- 
titiesy  which  are  affected  with  the  sign  — . 

17.  The  manner  in  which  we  generally  calculate  a  person's 
property,  is  a  good  illustration  of  what  bas  just  been  said.  We 
denote  what  a  man  really  possesses  by  positive  numbers,  using, 
or  understanding  the  sign  -f  5  whereas  his  debts  are  represent- 


Chap.  2.  Of  Simple  Quantities.  5 

ed  by  negative  numbers,  or  by  using  the  sign  — .  Thus,  when 
it  is  said  of  any  one  that  he  has  100  crowns,  but  owes  50,  this 
means  that  his  property  really  amounts  to  100  —  50  ;  or,  which 
is  the  same  thing,  -f  100  —  50,  that  is  to  say  50. 

18.  As  negative  numbers  may  be  considered  as  debts,  because 
positive  numbers  represent  real  possessions,  we  may  say  that 
negative  numbers  are  less  than  nothing.  Thus,  when  a  man 
has  nothing  in  the  world,  and  even  owes  50  crowns,  it  is  certain 
that  he  has  50  crowns  less  than  nothing ;  for  if  any  one  were  to 
make  him  a  present  of  50  crowns  to  pay  his  debts,  he  would 
still  be  only  at  the  point  nothing,  though  i^ally  richer  than 
before. 

19.  In  the  same  manner  therefore  as  positive  numbers  are 
incontestably  greater  than  nothing,  negative  numbers  are  less 
than  nothing.  Now  we  obtain  positive  numbers  by  adding  1  to 
0,  that  is  to  say,  to  nothing ;  and  by  continuing  always  to 
increase  thus  from  unity.  This  is  the  origin  of  the  series  of 
numbers  called  natural  numbers ;  the  following  are  the  leading 
terms  of  this  series : 

0,  +  1,  +  2,  +  3,  +  4,  +  5,  +  6,  +  7,  +  8,  +  9,  + 10, 
and  so  on  to  infinity. 

But  if  instead  of  continuing  this  series  by  successive  additions, 
we  continued  it  in  the  opposite  direction,  by  perpetually  sub- 
tracting unity,  we  should  have  the  series  of  negative  numbers  : 

0,  —  1,  —  2,  —  3,  ^  4,  —  5,  —  6,  —  7,  —  8,  -^  9,  —  10, 
and  so  on  to  infinity. 

20.  All  these  numbers,  whether  positive  or  negative,  have  the 
known  appellation  of  whole  numbers,  or  integers,  which  conse- 
quently are  eitlier  greater  or  less  than  nothing.  We  call  them 
integers,  to  distinguish  them  from  fractions,  and  from  several 
other  kinds  of  numbers  of  which  we  shall  hereafter  speak.  For 
instance,  50  being  greater  by  an  entire  unit  than  49,  it  is  easy 
to  comprehend  that  there  may  be  between  49  and  50  an  infinity 
of  intermediate  numbers,  all  greater  than  49,  and  yet  all  less 
than  50.  We  need  only  imagine  two  lines,  one  50  feet,  tlie 
other  49  feet  long,  and  it  is  evident  that  there  may  be  drawn  an 
infinite  number  of  lines  all  longer  than  49  feet,  and  yet  shorter 
than  50. 


&  Mgehra.  Sect.  1, 

21.  It  is  of  the  utmost  importance  throu8;h  the  whole  of 
Algebra,  that  a  precise  idea  be  formed  of  those  negative  quanti- 
ties about  whicli  we  liave  been  speaking.  I  shall  content  my- 
self with  remarking  here  that  all  such  expressions,  as 

+  1  —  1,  4-2  —  2,  +3  —3,  +4,-4,  &c. 
are  equal  to  0  or  nothing.     And  that 

+  2  —  5  is  equal  to  —  3  . 
For  if  a  person  has  2  crowns,  and  owes  5,  he  has  not  only 
nothing,  but  still  owes  S  crowns  :  in  the  same  manner 

7  —  12  is  equal  to  —  5,  and  25  —  40  is  equal  to  —  15. 

22.  The  same  observations  hold  true,  when,  to  make  the 
expression  more  general,  letters  are  used  instead  of  numbers  : 
0,  or  nothing  will  always  be  the  value  of  -f  a  —  a.  If  we  wish  to 
know  the  value  -fa  —  h  two  cases  are  to  be  considered. 

The  first  is  when  a  is  greater  than  h;  h  must  then  be  sub- 
tracted from  a,  and  the  remainder,  (before  winch  is  placed  or 
understood  to  be  placed  the  sign  -{-,)  shews  the  value  sought. 

Tlie  second  case  is  that  in  which  a  is  less  than  h  :  here  a  is 
to  be  subtracted  from  6,  and  the  remainder  being  made  negative, 
by  placing  before  it  the  sign  — ,  will  be  the  value  sought. 


CHAPTER  III. 

Of  the  Multiplication  of  Simple  Quantities, 

23.  When  there  are  two  or  more  equal  numbers  to  be  added 
together,  the  expression  of  their  sum  may  be  abridged ;  for 
example, 

a  +  a  is  the  same  with  2  x  a> 

a-fa-f-a  3Xflj 

a-fa-f-a-ffl  Ax  a,  and  so  on  ;  where  x  is  the  sign 

of  multiplication.  In  this  manner  we  may  form  an  idea  of  mul- 
tiplication ;  and  it  is  to  be  observed  that, 

2  X  a  signifies  2  times,  or  twice  a 

3  X  ft  3  times,  or  thrice  a 

4  X  a  4  times  a,  &c. 


Chap.  3.  Of  Simple  ^tantities.  f 

£4.  If  therefore  a  number  expressed  ly  a  letter  is  to  he  multiplied 
by  any  other  number,  we  simply  put  that  riumber  before  the  letter  ; 
thus, 

a  multipled  by  20  is  expressed  by  20a,  and 

b  multiplied  by  30  gives  30&,  &c. 

It  is  evident  also  that  c  taken  once,  or  Ic,  is  just  c. 

25.  Further,  it  is  extremely  easy  to  multiply  such  products 
again  by  other  numbers ;  for  example : 

2  times,  or  twice  3a  makes     6a, 

3  times,  or  thrice  4b  makes   12&, 
5  times  7x  makes  SScc, 

and  these  products  may  be  still  multiplied  by  pther  numbers  at 
pleasure. 

26.  When  the  number,  by  which  we  are  to  multiply,  is  also  re- 
presented by  a  letter,  we  place  it  immediately  before  the  other  letter  ^ 
thus,  in  multiplying  b  by  a,  the  product  is  written  ab ;  and  pq^ 
will  b6  the  product  of  the  multiplication  of  the  number  q  by  p> 
If  we  multiply  this  pq  again  by  a,  we  shall  obtain  apq, 

27.  It  may  be  remarked  here,  that  the  order  in  which  the  letters 
are  joined  together  is  indifferent ;  that  ab  is  the  same  thing  as  ba; 
for  b  multiplied  by  a  produces  as  much  as  a  multiplied  by  &. 
To  understand  this,  we  have  only  to  substitute  for  a  and  6 
known  numbers,  as  3  and  4 ;  and  the  truth  will  be  self-evident  i 
for  3  times  4  is  the  same  as  4  times  3. 

28.  It  will  not  be  difficult  to  perceive,  that  when  you  have  to 
put  numbers,  in  the  place  of  letters  joined  together,  as  we  have 
described,  they  cannot  be  written  in  the  same  manner  by  put- 
ting them  one  after  the  other.  For  if  we  were  to  write  34  for 
3  times  4,  we  should  have  34  and  not  12.  When  therefore  it  is 
required  to  multiply  common  numbers,  we  must  separate  them 
by  the  sign  x,  or  points  :  thus,  3x4,  or  3*4,  signifies  3  times  4, 
that  is  12.  So,  1  X  2  is  equal  to  2  ;  and  1x2x3  makes  6.  In 
like  manner  Ix2x3x4x  56  makes  1344 ;  and  1x2x3 
X4x5x6x7x8x9xl0is  equal  to  362^800,  &c. 

29.  In  tlie  same  manner,  we  may  discover  the  value  of  an 
expression  of  this,  form,  5*7*8'  abed.  It  shews  that  5  must  be 
multiplied  by  7,  and  that  this  product  is  to  be  again  multiplied 
by  8  ;  that  we  are  then  to  multiply  this  product  of  the  three 


Jlhehra.  Sect.  1* 


numbers  by  a,  next  by  6,  and  then  by  c,  and  lastly  by  d.  It  may 
be  observed  also,  that  instead  of  5  x  7  X  8  we  may  write  its  value, 
280;  for  we  obtain  this  number  when  we  multiply  the  product 
of  5  by  7,  or  35  by  8. 

SO.  The  results  which  arise  from  the  multiplication  of  two  or 
more  numbers  are  called  products  ;  and  the  numbers,  or  indivi- 
dual letters,  are  called  factors, 

31.  Hitherto  we  have  considered  only  positive  numbers,  and 
there  can  be  no  doubt,  but  that  the  products  which  we  have 
seen  arise  are  positive  also :  viz.  -}-  a  by  +  6  must  necessarily 
give  +  ab.  But  we  must  separately  examine  what  the  multi- 
plication of  +  a  by  . —  ft,  and  of  —  a  by  —  6,  will  produce. 

32.  Let  us  begin  by  multiplying  — a  by  3  or  -f- 3  ;  now  since 

—  a  may  be  considered  as  a  debt,  it  is  evident  that  if  we  take 
that  debt  three  times,  it  must  thus  become  three  times  greater, 
and  consequently  the  required  produ'^t  is  —  Sa.  So  if  we  multi- 
ply —  a  by  -f  &,  we  shall  obtain  —  ha^  or,  which  is  the  same  things 

—  ah.  Hence  we  conclude,  that  if  a  positive  quantity  be  multi- 
plied by  a  negative  quantity,  the  product  will  be  negative; 
and  lay  it  down  as  a  rule,  that  4-  by  +  makes  +,  or  pliis^  and 
that  on  the  contrary  +  by  — ,  or  —  by  +  gives  — ,  or  minus, 

33.  It  remains  to  resolve  the  case  in  which  —  is  multiplied  by 
— ;  or,  for  exami>le,  —  a  by  —  h.  It  is  evident^  at  first  sight, 
with  regard  to  the  letters,  that  the  product  will  be  ah;  but  it  is 
doubtful  whether  the  sign  -f-,  or  the  sign  — ,  is  to  be  placed 
before  the  product ;  all  we  know  is,  that  it  must  be  one  or  the 
other  of  these  signs.  Now  I  say  that  it  cannot  be  the  sign  —  : 
for  —  a  by  +  &  gives  —  a&,  and  —  a  by  —  h  cannot  produce  the 
same  result  as  —  a  by  +  6  ;  but  must  produce  a  contrary  result, 
that  is  to  say,  +  ah  ;  consequently  we  have  the  following  rule  ; 

—  multiplied  by  —  produces  +,  in  the  same  manner  as  +  mul- 
tiplied by  +.  [2.] 

34.  The  rules  which  we  have  explained  are  expressed  more 
briefly  as  follows  : 

Like  signs,  multiplied  together,  give  -f- ;  unlike  or  contrary  signs 
give  — .  Thus,  when  it  is  required  to  multiply  the  following 
numbers  ;  -fa,  —  h,  —  c,-\-d;  we  have  first  +  a  multiplied  by 

—  h,  which  makes—,  ah  ;  this  by  —  c,  gives  +  ahc  ;  and  this  by 
-f-  d,  gives  4-  ahcd. 


Chap.  4.  Of  Simple  Quantities.  9 

Q5.  The  difficulties  with  respect  to  the  signs  heing  removed, 
we  have  only  to  slievv  how  to  multiply  numbers  that  are  them- 
selves products.  If  we  were,  for  instance,  to  multiply  the 
number  ab  by  the  number  cd,  the  pi-oduct  w  ould  be  abed,  and  it 
is  obtained  by  multiplying  first  ab  by  c,  and  then  the  result  of 
that  multiplication  by  d.  Or,  if  we  had  to  multiply  36  by  12 ; 
since  12  is  equal  to  3  times  4,  we  should  only  multiply  36  first 
by  3,  and  then  the  product  108  by  4,  in  order  to  have  the  whole 
product  of  the  multiplication  of  12  by  36,  which  is  consequently 
432. 

36.  But  if  we  wished  to  multiply  5ah  by  3cd,  we  might  write 
Serf  X  5ab  ;  however,  as  in  the  present  instance  the  order  of  the 
numbers  to  be  multiplied  is  indifferent,  it  will  be  better,  as  is 
also  the  custom,  to  place  the  common  numbers  before  the  letters, 
and  to  express  the  product  thus  :  5  x  Sabcd,  or  I5abcd;  since  5 
times  3  is  15. 

So  if  we  had  to  multiply  12pqr  by  7xij,  we  should  oh  tain 
12  X  Tpqrxy,  or  S4pqrxy. 


CHAPTER  IV. 

Of  the  nature  of  whole  JK^umbers  or  Integers,  with  respect  to  their 

Factors. 

sr.  We  have  observed  that  a  product  is  generated  by  the 
multiplication  of  two  or  more  numbers  together,  and  that  these 
numbers  are  called  factors.  Thus  the  numbers  a,  b,  c,  d,  are 
the  factors  of  the  product  abed. 

38.  If,  therefore,  we  consider  all  whole  numbers  as  products 
of  two  or  more  numbers  multiplied  together,  we  shall  soon  find 
that  some  cannot  result  from  such  a  multiplication,  and  conse- 
quently have  not  any  factors  ;  wiiile  others  may  be  the  products 
of  two  or  more  multiplied  together,  and  may  consequently  have 
two  or  more  factors.  Thus,  4  is  produced  by  2  x  2  ;  6  by  2  x 
3  ,•  8  by  2  X  2  X  2  ;  or  27  by  3  X  3  x  3  ;  and  10  by  2  X  5,  &c. 

39.  But,  on  the  other  hand,  the  numbers,  2,  3,  5,  7,  11,  13, 
17,  &c.  cannot  be  represented  in  the  same  manner  by  fiictors, 
unless  for  that  pui^jose  we  make  use  of  unity,  and  represent  2, 

2 


10  Algebra.  Sect  1. 

for  instance,  by  1  x  2.  Now  the  numbers  which  are  multiplied 
by  1,  remaining  the  same,  it  is  not  proper  to  reckon  unity  as  a 
factor. 

All  numbers  therefore,  such  as  2,  3,  5,  7,  11,  13,  17,  &c. 
which  cannot  be  represented  by  factors,  are  called  simpUf  or 
prime  numbers  ;  whereas  others,  as  4.  6,  8,  9,  10,  12,  14,  15,  16^ 
18,  &c,  which  may  be  represented  by  factors,  are  called  com- 
pound  numbers, 

40.  Simple  or  prime  numbers  deserve  therefore  particular 
attention,  since  they  do  not  result  from  the  multiplication  of  two 
or  more  numbers.  It  is  particularly  worthy  of  observation, 
that  if  we  write  these  numbei's  in  succession  as  they  follow 
each  other,  tlms ; 

2,  3,  5,  7,  11,  13,  17,  19,  23,  29,  31,  37,41,  43,  47,  &c.  [3.] 
we  can  trace  no  regular  order ;  their  increments  are  sometimes 
greater,  sometimes  less ;  and  hitherto  no  one  has  been  able  to 
discover  whether  they  follow  any  certain  law  or  not. 

41.  Ml  compound  numbers,  which  may  be  represented  by  factors , 
result  from  the  prime  numbers  above  mentioned  ;  that  is  to  say,  all 
their  factors  are  prime  numbers.  For,  if  we  find  a  factor  which 
is  not  a  prime  number,  it  may  always  be  decomposed  and  repre- 
sented by  two  or  more  prime  numbers.  When  we  have  repre- 
sented, for  instance,  the  number  30  hy  5  x  6,  it  is  evident  that  6 
not  being  a  prime  number,  but  being  produced  by  2  x  3,  we 
might  have  represented  30  by  5  x  2  x  3,  or  by  2  x  3  x  5  ;  that 
is  to  say,  by  factors,  which  are  all  prime  numbers. 

42.  If  we  now  consider  those  compound  numbers  which  may 
be  resolved  into  prime  numbers,  we  shall  observe  a  great  differ- 
ence among  them  ;  we  shall  find  that  some  have  only  two  factors, 
that  others  have  three,  and  others  a  still  greater  number.  We 
have  already  seen,  for  example,  that 


4  is  the  same  as  2x2, 

8  2X2X2, 

10  2X5, 

14  2X7, 

16  2X2X2X2, 

43.  Hence  it  is  easy  to  find  a  method  for  analysing  any  num- 
ber, or  resolving  it  into  its  simple  factors.     Let  there  be  pro- 


6  is  the  same  as  2x5, 

9  3XS, 

12  2x3x2, 

15  3X5, 

and  so  on. 


Chap.  5.  Of  Simple  ^antities,  11 


posed,  for  instance,  the  number  360  ;  we  shall  represent  it  first 
by  2  X  180.     Now  180  is  equal  to  2  x  90,  and 
90-)  r2x45, 

45  I  is  the  same  as  -<  3x15,  and  lastly 
15J  (.3X5. 

So  that  the  number  360  may  be  represented  by  these  simple 
factors,  2x2x2x3x3x5;  since  all  these  numbers  multiplied 
together  produce  360.  [4.] 

44.  This  shews,  that  the  prime  numbers  cannot  be  divided  by 
other  numbers,  and  on  the  other  hand,  that  the  simple  factors  of' 
compound  numbers  are  found,  most  conveniently  and  with  the 
greatest  certainty,  by  seeking  the  simple,  or  prime  numbers,  by 
which  those  compound  numbers  are  divisible.  But  for  this  division 
is  necessary  ;  we  shall  therefore  explain  the  rules  of  that  opera* 
tion  in  the  following  chapter. 


CHAPTER  V. 

Of  the  Division  of  Simple  Quantities, 

45.  When  a  number  is  to  be  separated  into  two,  three,  or 
more  equal  parts,  it  is  done  by  means  o?  division,  which  enables 
us  to  determine  the  magnitude  of  one  of  those  parts.  When  we 
wish,  for  example,  to  separate  the  number  12  into  three  equal 
parts,  we  find  by  division  that  each  of  those  parts  is  equal  to  4. 

The  following  terms  are  made  use  of  in  this  operation.  The 
number,  which  is  to  be  decompounded  or  divided,  is  called  the 
dividend  ;  tlic  number  of  equal  parts  sought  is  called  the  divisor; 
the  magnitude  of  one  of  those  parts,  determined  by  the  division, 
is  called  the  quotient :  thus,  in  the  above  example ; 
12  is  the  dividend, 

3  is  tlie  divisor,  and 

4  is  the  quotient. 

46.  It  follows  from  this,  that  if  we  divide  a  number  by  2,  or 
into  two  equal  parts,  one  of  those  paints,  or  the  quotient,  taken 
twice,  makes  exactly  the  number  proposed ;  and,  in  the  same 
manner,  if  we  have  a  number  to  be  divided  by  3,  the  quotient 
taken  thrice  must  give  the  same  number  again.    In  general,  the 


12  Mgebra.  Sect.  1. 

multiplication  of  the  qmtient  bij  the  divisor  must  always  reproduce 
the  dividend, 

47.  It  is  for  this  reason  that  division  is  said  to  be  a  rule, 
which  teaches  us  to  find  a  number  or  quotient,  whicli,  being 
multiplied  by  the  divisor,  will  exactly  produce  the  dividend. 
For  example,  if  35  is  to  be  divided  by  5,  we  seek  for  a  number 
which,  multiplied  by  5,  will  produce  55,  Now  this  number  is 
7,  since  5  times  7  is  35,  The  manner  of  expression,  em- 
ployed in  this  reasoning,  is  ;  5  in  35,  7  times  ;  and  5  times  7 
makes  35. 

48.  The  dividend  therefore  may  be  considered  as  a  product, 
of  which  one  of  the  factors  is  the  divisor,  and  the  other  the 
quotient.  Thus,  supposing  we  have  63  to  divide  by  7,  we  en- 
deavour to  find  such  a  product,  that  taking  7  for  one  of  its 
factors,  the  other  factor  multiplied  by  this  may  exactly  give  63. 
Now  7  X  9  is  such  a  product,  and  consequently  9  is  the  quotient 
obtained  when  we  divide  63  by  7. 

49.  In  general,  if  we  have  to  divide  a  number  ab  by  a,  it  is 
evident  that  the  quotient  will  be  b  ;  for  a  multiplied  by  b  gives 
the  dividend  ab.  It  is  clear  also,  that  if  we  had  to  divide  ab  by 
h,  the  quotient  would  be  a.  And  in  all  examples  of  division 
that  can  be  proposed,  if  we  divide  the  dividend  by  the  quotient, 
we  shall  again  obtain  the  divisor ;  for  as  24  divided  by  4  gives 
6,  so  24  divided  by  6  will  give  4. 

50.  As  the  whole  opei'ation  consists  in  representing  the  dividend 
by  two  factors,  of  which  one  may  be  equal  to  the  divisor,  the  other 
to  the  quotient ;  the  following  examples  will  be  easily  understood. 
I  say  first,  that  the  dividend  abc,  divided  by  a,  gives  be ;  for  a 
multiplied  by  be,  produces  abc :  in  the  same  manner  abc,  being 
divided  by  b,  we  shall  have  ac  ;  and  abc,  divided  by  ac,  gives  b, 
I  say  also,  that  12mn^  divided  by  3m,  gives  4n;  for  5m,  multi- 
plied by  4 n,  makes  12mn,  But  if  this  same  number  12mii  had 
been  divided  by  12,  we  should  have  obtained  the  quotient  mn. 

51.  Since  every  number  a  may  be  expressed  by  la  or  one  a,  it 
is  evident  that  if  we  had  to  divide  a  or  la  by  1,  the  quotient 
would  be  the  same  number  a.  But,  on  the  contrary,  if  the  same 
number  a,  or  la  is  to  be  divided  by  a,  the  quotient  will  be  1. 


Chap.  5.  Of  Simple  (Quantities.  li 

52.  It  often  happens  that  we  cannot  represent  the  dividend  as 
the  product  of  two  factors,  of  w  hich  one  is  equal  to  the  divisor ; 
and  then  the  division  cannot  be  performed  in  the  manner  we 
have  described. 

When  we  have,  for  example,  24  to  be  divided  by  7,  it  is  at 
first  sight  obvious,  that  tlie  number  7  is  not  a  factor  of  24  ;  for 
the  product  of  7  X  3  is  only  21,  and  consequently  too  small,  and 
7x4  makes  28,  which  is  greater  than  24.  We  discover  however 
from  this,  that  the  quotient  must  be  greater  than  3,  and  less  than 
4.  In  order  therefore  to  determine  it  exactly,  we  employ  anotlier 
species  of  numbers,  which  are  called  fractions^  and  which  we 
shall  consider  in  one  of  the  following  chapters. 

53.  Until  the  use  of  fractions  is  considered,  it  is  usual  to  rest 
satisfied  with  the  whole  number  which  approaches  nearest  to 
the  true  quotient,  but  at  the  same  time  paying  attention  to  the 
remainder  which  is  left ;  thus  we  say,  7  in  24,  3  times,  and  the 
remainder  is  3,  because  3  times  7  produces  only  21,  which  is  3 
less  than  24.  We  may  consider  the  following  examples  in  the 
same  manner : 

6)34(5,    that  is  to  say  the  divisor  is  6,  the  dividend  34, 
30         the  quotient  5,  and  the  remainder  4. 

4 
9)41(4,     here  the  divisor  is  9,  the  dividend  41,  the  quo- 
36         tient  4,  and  the  remainder  5. 


The  following  rule  is  to  be  observed  in  examples  where  there 
is  a  remainder. 

54.  Ijxve  multiply  the  divisor  by  the  quotient,  and  to  the  product 
add  the  remainder,  we  must  obtain  the  dividend;  this  is  the 
method  of  proving  the  division,  and  of  discovering  whether  the 
calculation  is  right  or  not.  Thus,  in  the  first  of  the  two  last 
examples,  if  we  multiply  6  by  5,  and  to  the  product  30  add  the 
remainder  4,  we  obtain  34,  or  the  dividend.  And  in  the  last 
example,  if  we  multiply  the  divisor  9  by  the  quotient  4,  and  to 
the  product  36  add  the  remainder  5,  we  obtain  the  dividend  41. 


li  Mgebra.  Sect.  1. 

55.  Lastly,  it  is  necessary  to  remark  here,  with  regard  to  the 
signs  -f-  plus  and  —  minus,  that  if  we  divide  -f  a&  by  4-  a,  the 
quotient  will  be  -f-  ft,  which  is  evident.  But  if  we  divide  -}-  ah 
by  —  ttf  the  quotient  will  be  —  b  ;  because — a  x  — b  gives  -f  ab. 
If  the  dividend  is  —  aft,  and  is  to  be  divided  by  the  divisor  -f  a, 
the  quotient  will  be  —  b;  because  it  is  < —  b,  which,  multiplied 
by  +  a,  makes  —  ab.  Lastly,  if  we  have  to  divide  the  dividend 
—  ab  by  the  divisor  —  a,  the  quotient  will  be  +  &  ;  for  the  divi- 
dend —  ab  is  the  product  of  —  a  by  +  6. 

56.  With  regard  therefore  to  the  signs  -f-  and  — ,  division  admits 
the  same  rides  that  we  have  seen  applied  in  multiplication  ;  viz. 

4-  by  4-  requires  -f- ;  -f  by  —  requires  — ; 

—  by  -f-  requires  — ;  —  by  —  requires  -f-  : 

or  in  a  few  words,  like  signs  give  plus,  unlike  signs  give  minus, 

57.  Thus,  when  we  divide  ISpq  by  —  Sjh  the  quotient  is  —  6q. 
Farther ; 

—  SOxy  divided  by  -f-  6y  gives  —  5^,  and 

—  54abc  divided  by  —  96  gives  -|-  6ac  ; 

for  in  this  last  example,  —  96  multiplied  by  +  6ac  makes  —  6x 
9aJbc,  or  —  54abc.     But  we  have  said  enough  on  the  division  of 
simple  quantities  ;  we  shall  therefore  hasten  to  the  explanation 
of  fractions,  after  having  added  some  farther  remarks  on  the 
nature  of  numbers,  with  respect  to  their  divisors. 


CHAPTER  VI. 

Of  the  properties  of  Integers  with  respect  to  their  Divisors. 

58.  As  we  have  seen  that  some  numbers  are  divisible  by  cer- 
tain divisors,  while  others  ar«  not  ;  in  order  that  we  may 
obtain  a  more  particular  knowledge  of  numbers,  this  difference 
must  be  carefully  observed,  both  by  distinguishing  the  numbers 
that  are  divisible  by  divisors  from  those  which  arc  not,  and  by 
considering  the  remainder  that  is  left  in  the  division  of  the 
latter.     For  this  purpose  let  us  examine  the  divisors  ; 

2,  3,  4,  5,  6,  r,  8,  9,  10,  kc. 

59.  First,  let  the  divisor  be  2 ;  the  numbers  divisible  by  it 
are,  2,4,  6,  8,  10,  12,  14,  16,  18,  20,  &c.  which,  it  appears 


Chap.  6,  Of  Simple  Quantities,  19 

increase  always  by  two.  These  numbers,  as  far  as  they  can  be 
continued,  are  called  even  numbers.  But  there  are  other  num* 
bers,  viz, 

1,  3,  5,  r,  9,  11,  13,  15,  17,  19,  kc. 
which  are  uniformly  less  or  greater  than  the  former  by  unity, 
and  which  cannot  be  divided  by  2,  without  the  remainder  1 ; 
these  are  called  odd  numbers. 

The  even  numbers  are  all  comprehended  in  the  general  expres- 
sion 2fl  ;  for  they  are  all  obtained  by  successively  substituting 
for  a  the  integers  1,  2,  3,  4,  5,  6,  7,  &c.  and  hence  it  follows  that 
the  odd  numbers  are  all  comprehended  in  the  expression  2a  -{- 1 
because  2a  +  1  is  greater  by  unity  than  the  even  number  2a. 

60.  In  the  second  place,  let  the  number  3  be  the  divisor ;  the 
numbers  divisible  by  it  are, 

3,  6,  9,  12,  15,  18,  21,  24,  27,  30,  and  so  on  ;  and  these  num- 
bers may  be  represented  by  the  expression  3a  ,•  for  3a  divided 
by  3  gives  the  quotient  a  without  a  remainder.  All  other  num- 
bers, which  we  would  divide  by  3,  will  give  1  or  2  for  a  remain- 
der, and  are  consequently  of  two  kinds.  Those,  which  after  the 
division  leave  tlie  remainder  1,  are ; 

1,4,7,10,13,  16,19,  &c. 
and  are  contained  in  the  expression  3a  -f  1  ^  but  the  other  kind, 
where  the  numbers  give  the  remainder  2,  are ; 
2,5,8,11,  14,17,20,  &c, 
and  they  may  be  generally  expressed  by  Sa  -f  2;  so  that  all 
numbers  may  be  expressed  either  by  3a,  or  by  3a  -f-  1,  or  by 
3a -f- 2. 

61.  Let  us  now  suppose  that  4  is  the  divisor  under  considera- 
tion :  the  numbers  which  it  divides  are  ; 

4,  8,  12,  16,  20,24,  kc, 
which  increase  uniformly  by  4,  and  are  comprehended  in  the 
expression  4a.     All  other  numbers,  that  is,  those  which  are  not 
divisible  by  4,  may  leave  the  remainder  1,  or  be  greater  than 
the  former  by  1  :  as 

1,5,9,  13,  17,  21,25,  &C. 
and  consequently  may  be  comprehended  in  the  expression  4a  -f- 
1 :  or  they  may  give  the  remainder  2  ;  as 

2,  6,  10,  14,  18,  22,  26,  &c. 


te  Mgehra.  S^ct.  U 


"d 


and  be  expressed  by  4a  +  2 ;  or,  lastly,  they  may  give  the 
remainder  3 ;    as 

3,  7,  11,  15,  19,  23,  27,  &,c. 
and  may  be  rejjresented  by  the  expression  4a  -f-  3. 

AH  possible  integral  numbers  are  therefore  contained  in  one  or 
other  of  these  four  expressions  ; 

4a,  4a  4-  1,  4a  +  2,  4a  +  3. 

62.  It  is  nearly  the  same  when  the  divisor  is  5  ;  for  all  num- 
bers which  can  be  divided  by  it  are  comprehended  in  the 
expression  Sa,  and  those  which  cannot  be  divided  by  5,  ai'e 
reducible  to  one  of  the  following  expressions  : 

5a  -f  1,  5a  +  2,  5a  +  3,  5a  -f  4  ; 
and  we  may  go  on  in  the  same  manner  and  consider  the  great- 
/Cst  divisors. 

63.  It  is  proper  to  recollect  here  what  has  been  already  said 
on  the  resolution  of  numbers  into  their  simple  factors ;  for  every 
number,  among  the  factors  of  which  is  found, 

2,  or  3,  or  4,  or  5,  or  7, 
or  any  other  number,  will  be  divisible  by  those  numbers.     For 
example ;  60  being  equal  to  2  x  2  x  3  x  5,  it  is  evident  that  60 
is  divisible  by  2,  and  by  3,  and  by  5.  [5.] 

64.  Farther,  as  the  general  expression  ahcd  is  not  only  divi- 
sible by  a,  and  6,  and  c,  and  d,  but  also  by 

abf  ac,  ad,  he,  bd,  cd,  and  by 

abCf  abd,  acd,  bed,  and  lastly  by 

abed,  that  is  to  say,  its  own  value; 
it  follows  that  60,  or  2  x  2  x  3  x  5,  may  be  divided  not  only  by 
these  simple  numbers,  but  also  by  those  which  are  composed  of 
two  of  them  ;  that  is  to  say,  by  4,  6,  10,  15  :  and  also  by  those 
which  are  composed  of  three  simple  factors,  that  is  to  say,  by 
12,  20,  30,  and  lastly  also,  by  60  itself. 

65.  When,  therefore,  we  have  represented  any  number,  assumed 
at  pleasure,  by  its  simple  factors,  it  will  be  very  easy  to  shew  all 
the  numbers  by  which  it  is  divisible.  For  we  luive  only,  first,  to 
take  the  simple  factors  one  by  one,  and  then  to  multiply  them  togeth- 
er two  by  two,  three  by  three,  four  by  four,  ^c.  till  we  arrive  at 
the  number  proposed. 

66.  It  must  here  be  particularly  observed,  that  every  number 
is  divisible  by  1 ;  and  also  that  every  number  is  divisible  by 


Chap.  7.  Of  Simple  quantities.  Vf 

itself;  so  that  every  nmnher  lias  at  least  two  factors,  or  divisors, 
the  number  itself  and  unity  :  but  every  number  which  has  no 
other  divisor  than  these  two,  belongs  to  the  class  of  numbers, 
which  we  have  before  called  simjjle,  or  prime  numbers. 

Except  these,  all  other  numbers  have,  beside  unity  and  them- 
selves, other  divisors,  as  may  be  seen  from  the  following  table, 
in  w  hich  are  placed  under  each  number  all  its  divisors.[6.] 

TABLE. 


1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

2 

3 

2 

5 

2 

7 

2 

3 

2 

11 

2 

13 

2 

3 

2 

17 

2 

19 

2 

4 

3 
6 

4 

8 

9 

5 
10 

3 

4 

6 

12 

7 
14 

5 
15 

4 

8 
16 

3 
6 
9 

18 

4 

5 

10 

20 

1 

2 

2 

3 

2 

4 

2 

4 

5 

4 

2 

6 

2 

4 

4 

5 

2 

6 

2 

6 

p. 

P. 

P. 

P. 

P. 

P. 

P. 

P. 

P. 

67.  Lastly,  it  ought  to  be  observed  that  0,  or  nothing,  may  be 
considered  as  a  number  which  has  the  property  of  being  divisi- 
ble by  all  possible  numbers ;  because  by  whatever  number  a 
we  divide  0,  the  quotient  is  always  0  ;  for  it  must  be  remarked 
that  the  multiplication  of  any  number  by  nothing  produces 
nothing,  and  therefore  0  times  a,  or  Oa,  is  0. 


CHAPTER  VII. 

Of  Fractions  in  general, 

68.  When  a  number,  as  7,  for  instance,  is  said  not  to  be 

divisible  by  another  number,  let  us  suppose  by   3,  this   only 

means,  that  the  quotient  cannot  be  expressed  by  an  integral 

number ;  and  it  must  not  be  thought  by  any  means  that  it  is 

5 


18  Mgebra,  Sect.  1. 

impossible  to  form  an  idea  of  that  quotient.  Only  imagine  a 
line  of  7  feet  in  length,  nobody  can  doubt  the  possibility  of 
dividing  this  line  into  3  equal  parts,  and  of  forming  a  notion  of 
the  length  of  one  of  those  parts. 

69.  Since  therefore  we  may  form  a  precise  idea  of  the  quo- 
tient obtained  in  similar  cases,  though  that  quotient  is  not  an 
integral  number,  this  leads  us  to  consider  a  particular  species  of 
numbers,  called  fractions,  or  broken  numbers.  The  instance 
adduced  furnishes  an  illustration.  If  we  have  to  divide  7  by  3, 
we  easily  conceive  the  quotient  which  should  result,  and  express 
it  by  I ;  placing  the  divisor  under  the  dividend,  and  separating 
the  two  numbers  by  a  stroke,  or  line. 

70.  So,  in  general,  when  the  number  a  is  to  be  divided  by  the 
number  b,  we  represent  the  quotient  by  -|,  and  call  this  form  of 
expression  a  fraction.  We  cannot  therefore  give  a  better  idea  of 
a  fraction  -|^,  than  by  saying  that  we  thus  express  the  quotient 
resulting  from  the  division  of  the  upper  number  by  the  lower. 
We  must  remember  also,  that  in  all  fractions  tlie  lower  num- 
ber is  called  the  denominator,  and  that  above  the  line  the  nume- 
rator, 

71.  In  the  above  fraction,  -J,  which  we  read  seven  thirds,  7  is 
the  numerator,  and  3  tlie  denominator.  We  must  also  read  |,  two 
thirds  ;  |,  three  fourths  ;  4,  three  eighths ;  J^*^,  twelve  hun- 
dredths ;  and  -J,  one  half. 

72.  In  order  to  obtain  a  more  perfect  knowledge  of  the 
nature  of  fractions,  we  shall  begin  by  considering  the  case  in 
which  the  numerator  is  equal  to  the  denominator,  as  in  — . 
Now,  since  this  expresses  the  quotient  obtained  by  dividing  a 
by  a,  It  is  evident  that  this  quotient  is  exactly  unity,  and  that 
consequently  this  fraction  —  is  equal  to  ] ,  or  one  integer ;  for 
the  same  reason,  all  the  following  fractions, 

2       3      4*678      Rrp 
■5»  T'   4»  T»   6-'  T'  ¥>  ^^' 

are  equal  to  one  another,  each  being  equal  to  1,  or  one  integer. 

73.  We  have  seen  that  a  fraction,  whose  numerator  is  equal  to 
tlie  denominator,  is  equal  to  unity.  All  fractions  therefore 
wliosc  numerators  arc  less  than  the  denominators,  have  a  value 


Chap.  7.  Of  Simple  Quantities.  19 

less  than  unity.  For  if  I  liave  a  number  to  be  divided  by  ano- 
ther which  is  greater,  the  result  must  necessarily  be  less  than  1 ; 
if  we  cut  a  line,  for  example,  two  feet  long,  into  three  parts,  one 
of  those  parts  will  undeniably  be  shorter  than  a  foot :  it  is 
evident  then,  that  |  is  less  than  1,  for  the  same  reason,  that  the 
numerator  2  is  less  than  the  denominator  3. 

74.  If  the  numerator,  on  the  contrary,  be  greater  than  the 
denominator,  the  value  of  the  fraction  is  greater  than  unity. 
Thus  I  is  greater  than  1,  for  |  is  equal  to  |  together  with  |. 
Now  I  is  exactly  1,  consequently  |  is  equal  to  1  +  |,  that  is,  to 
an  integer  and  a  half.  In  the  same  manner  ^  is  equal  to  J-|,  4 
to  If,  and  |-  to  2^.  And  in  general,  it  is  sufficient  in  such  cases 
to  divide  the  upper  number  by  the  lower,  and  to  add  to  the 
quotient  a  fraction  having  the  remainder  for  the  numerator,  and 
the  divisor  for  the  denominator.  If  the  given  fraction  were,  for 
example,  :}|,  we  should  have  for  the  quotient  3,  and  7  for  the 
remainder  ;  whence  we  should  conclude  that  4^  is  the  same  as 

73,  Thus  we  see  how  fractions,  whose  numerators  are  greater 
than  the  denominators,  are  resolved  into  two  members ;  one  of 
which  is  an  integer,  and  the  other  a  fractional  number,  having 
the  numerator  less  than  the  denominator.  Such  fractions  as 
contain  one  or  niore  integers,-  are  called  improper  fractions^  to 
distinguish  them  from  fractions  properly  so  called,  which,  having 
the  numerator  less  than  the  denominator,  are  less  than  unity,  or 
than  an  integer. 

76.  The  nature  of  fractions  is  frequently  considered  in  an- 
other way,  which  may  throw  additional  light  on  the  subject. 
If  we  consider,  for  example,  the  fraction  |,  it  is  evident  that  it 
is  three  times  greater  than  -J.  Now  this  fraction  J  means,  that 
if  we  divide  1  into  4  equal  parts,  this  will  be  the  value  of  one  of 
those  parts  ;  it  is  obvious  then,  that  by  taking  3  of  those  parts, 
we  shall  have  the  value  of  the  fraction  |. 

In  the  same  manner  we  may  consider  every  other  fraction  ; 
for  example,  -J^  ;  if  we  divide  unity  into  12  equal  parts,  7  of 
those  parts  will  be  equal  to  the  fraction  proposed. 

77.  From  this  manner  of  considering  fractions,  tlie  expres- 
sions numerator  and  denominator  arc  derived.     For,  as  in  the 


26  Mgehra.  Sect.  1. 

preceding  fraction  -^^^  the  number  under  the  line  shews  that  13 
is  the  number  of  parts  into  which  unity  is  to  be  divided ;  and  as 
it  may  be  said  to  denote,  or  name  the  parts,  it  has  not  impro- 
perly been  called  the  deiwininator. 

Farther,  as  the  upper  number,  viz.  7,  shews  that,  in  order  to 
have  the  value  of  the  fraction,  we  must  take,  or  collect  7  of 
those  parts,  and  therefore  may  be  said  to  reckon,  or  number 
them,  it  has  been  thought  proper  to  call  the  number  above  the 
line  the  numerator. 

78.  As  it  is  easy  to  understand  what  |  is,  when  we  know  the 
signification  of  ^,  we  may  consider  the  fractions  whose  nume- 
rator is  unity,  as  the  foundation  of  all  others.  Such  are  the 
fractions, 

iiiiiiiii      1      1     A-f 

"5'  7'   4'  T'  r^  T>  ¥»  T^  T^»  IT'  Tf '    *^^* 

and  it  is  observable  that  these  fractions  go  on  continually  dimin- 
ishing :  for  the  more  you  divide  an  integer,  or  the  greater  the 
number  of  parts  into  which  you  distribute  it,  the  less  does  each 
of  those  parts  become.  Thus  ^J^  is  less  than  Jg^ ;  ^/^^  is  less 
than  ^1^  ;  and  ^^J^^  is  less  than  ^^Vtt- 

79.  As  we  have  seen,  that  the  more  we  increase  the  denomi- 
nator of  such  fractions,  the  less  their  values  become  ;  it  may  be 
asked,  whether  it  is  not  possible  to  make  the  denominator  so 
great,  that  the  fraction  shall  be  reduced  to  nothing  ?  I  answer, 
no  ;  for  into  whatever  number  of  parts  unity  (the  length 
of  a  foot  for  instance)  is  divided ;  let  those  parts  be  ever  so 
small,  they  will  still  preserve  a  certain  maeruitude,  and  there- 
fore can  never  be  absolutely  reduced  to  nothing. 

80.  It  is  true,  if  we  divide  the  length  of  a  foot  into  1000  parts ; 
those  parts  will  not  easily  fall  under  the  cognizance  of  our 
senses  :  but  view  them  through  a  good  microscope,  and  each  of 
them  will  appear  large  enough  to  be  subdivided  into  100  parts, 
and  more. 

At  present,  however,  we  have  nothing  to  do  with  what  de- 
pends on  ourselves,  or  vy^ith  what  we  are  capable  of  performing, 
and  what  our  eyes  can  perceive ;  the  question  is  rather,  what  is 
possible  in  itself.  And,  in  this  sense  of  the  word,  it  is  certain, 
that  however  great  we  suppose  the  denominator,  the  fraction 
will  never  entirely  vanish,  or  become  equal  to  0. 


Chap.  7.  Of  Simple  ^antities.  21 

81.  We  never  therefore  arrive  completely  at  nothing,  how- 
ever great  the  deuominator  may  be  ;  and  those  fractions  always 
preserving  a  certain  quantity,  we  may  continue  the  series  of 
fractions  in  the  78th  article  without  interruption.  This  circum- 
stance has  introduced  the  expression,  that  the  denominator  must 
be  infinite,  or  infinitely  great,  in  order  that  the  fraction  may  be 
reduced  to  0,  or  to  nothing ;  and  the  word  infinite  in  reality 
signifies  here,  that  we  should  never  arrive  at  the  end  of  the 
series  of  the  above  mentioned  fractions. 

82.  To  express  this  idea,  which  is  extremely  well  founded, 
we  make  use  of  the  sign  oo  ,  which  consequently  indicates  a 
number  infinitely  great ;  and  we  may  therefore  say  that  this 
fraction  i  is  really  nothing,  for  the  very  reason  that  a  fraction 
cannot  be  reduced  to  nothing,  until  the  denominator  has  been 
increased  to  infinity. 

83.  It  is  the  more  necessary  to  pay  attention  to  this  idea  of 
infinity,  as  it  is  derived  from  the  first  foundations  of  our  know- 
ledge, and  as  it  will  be  of  the  greatest  importance  in  the  follow- 
ing part  of  this  treatise. 

We  may  here  deduce  from  it  a  few  consequences,  that  are 
extremely  curious  and  worthy  of  attention.  The  fraction  ^ 
represents  the  quotient  resulting  from  the  division  of  the  divi- 
dend 1  by  the  divisor  oo.  Now  we  know  that  if  we  divide 
the  dividend  1  by  the  quotient  J,  which  is  equal  to  0,  we  obtain 
again  the  divisor  oo  :  hence  we  acquire  a  new  idea  of  infinity  ; 
we  learn  that  it  arises  from  the  division  of  1  by  0  ;  and  we  are 
therefore  entitled  to  say,  that  1  divided  by  0  expresses  a  number 
infinitely  great,  or  oo . 

84.  It  may  be  necessary  also  in  this  place  to  correct  the 
mistake  of  those  who  assert,  that  a  number  infinitely  great  is 
not  susceptible  of  increase.  This  opinion  is  inconsistent  with 
the  just  principles  which  we  have  laid  down  ;  for  J  signifying  a 
number  infinitely  great,  and  ^  being  incontestably  the  double  of 
^9  it  is  evident  tha^*  a  number,  though  infinitely  great,  may  still 
become  two  or  more  times  greater.  [7.] 


id  Algebra.  Sect.  1. 

CHAPTER.  VIII. 

Of  the  propei'ties  of  Fractions, 

85.  We  have  already  seen,  that  each  of  the  fractions, 

2       3      4      5       6       7       8      ;Brp 
¥'  T'  "i*  T'  7'  T'  "S"'  "''^* 

makes  an  integer,  and  that  consequently  they  are  all  equal  to 
one  another.  The  same  equality  exists  in  the  following  frac- 
tions, 

2      46       8       10       13      Xrr 

each  of  them  making  two  integers ;  for  the  numerator  of  each, 
divided  by  its  denominator,  gives  2.     So  all  the  fractions 

3      6       9       13       15       18      X^r 

T'  5^'  7'    "S  9    J  '    T  '  ^^' 

are  equal  to  one  another,  since  3  is  their  common  value. 

86.  We  may  likewise  represent  the  value  of  any  fraction,  in 
an  infinite  variety  of  ways.  For  if  we  multiply  both  the  nume- 
rator and  the  denominator  of  a  fraction  by  the  same  number,  which 
may  be  assumed  at  pleasure,  this  fraction  will  still  preserve  the 
same  value.    For  this  reason  all  the  fractions 

12S45  6  7  8  9         10      ArP 

¥'  4>  y»  T'  TIT'  Tf '  T4'  TT'  TT'  ^TT'  *^^' 

are  equal,  the  value  of  each  being  |.     Also 

1334  5  6  7  8  910      Jirr 

7'  T'  7'  TJ'  TT»  T7'  fT'  SS*  aT'  TSlf'  *^^* 

are  equal  fractions,  the  value  of  each  of  which  is  ^.  The  frac- 
tions 

24        8        10      12       1416      Arp 
S9    69  1^9  f-g9  T¥'   ^T'   Si:9   ^^' 

have  likewise  all  the  same  value :  and  lastly,  we  may  conclude 
in  general,  that  the  fraction  ^  may  be  represented  by  the  fol- 
lowing expressions,  each  of  which  is  equal  to  -|^ ;  viz. 

a    2a    Sa   4a    5a    6a   7a' 
T'  2b'  W  46'  5&'  66'  76'  ^*'- 

87.  To  be  convinced  of  this  we  have  only  to  write  for  the 
value  of  the  fraction -^  a  certain  letter  c,  representing  by  this 
letter  c  the  quotient  of  the  division  of  a  by  6  ;  and  to  recollect 
tbat  the  multiplication  of  the  quotient  c  by  the  divisor  6,  must  give 
the  dividend.  For  since  c  multiplied  by  b  gives  a,  it  is  evident  that 
c  multiplied  by  26  will  give  2a,  that  c  multiplied  by  36  will  give 


Chap.  8.  Of  Simple  Quantities.  sa 

Sa,  and  that  in  general  c  multiplied  by  mb  must  give  ma.  Now 
changing  this  into  an  example  of  division,  and  dividing  the  pro- 
duct ma  by  mb,  one  of  the  factors,  the  quotient  must  be  equal  to 
the  other  factor  c  ;  but  ma  divided  by  mb  gives  also  the  fraction 
-^,  which  is  consequently  equal  to  c  ;  and  this  is  what  was  to 

mb 

be  proved  :  for  c  having  been  assumed  as  the  value  of  the  frac- 
tion   A  it  is  evident  that  this  fraction  is  equal  to  the  fraction 

^,  whatever  be  the  value  of  m, 

mo 

88.  We  have  seen  that  every  fraction  may  be  represented  in  an 
infinite  number  of  forms,  each  of  which  contains  the  same  value ; 
and  it  is  evident  that  of  all  these  forms,  that  which  shall  be 
composed  of  the  least  numbers,  will  be  most  easily  understood. 
For  example,  we  might  substitute  instead  of  |  the  following 
fractions, 

6'  T'  T5'  TT'  T"5''  *^^' 

but  of  all  these  expressions  |  is  that  of  which  it  is  easiest  to 
form  an  idea.  Here  therefore  a  problem  arises,  how  a  fraction, 
such  as  ^-j,  which  is  not  expressed  by  the  least  possible  numbers, 
may  be  reduced  to  its  simplest  form,  or  to  its  least  terms,  that  is 
to  say,  in  our  present  example,  to  |. 

89.  It  will  be  easy  to  resolve  this  problem,  if  we  consider  that 
a  fraction  still  preserves  its  value,  when  we  multiply  both  its 
terms,  or  its  numerator  and  denominator,  by  the  same  number* 
For  from  this  it  follows  also,  that  if  we  divide  the  numerator  and 
denominator  of  a  fraction  by  the  same  number,  the  fraction  still 
preserves  the  same  value.  This  is  made  more  evident  by  means 
of  tlie  general  expression  ^ ;  for  if  we  divide  both  the  nume- 
rator ma  and  the  denominator  mb  by  the  number  m,  we  obtain 
the  fraction  -^,  which,  as  was  before  proved,  is  equal  to  ^. 

o  mo 

90.  In  order  therefore  to  reduce  a  given  fraction  to  its  least 
terms,  it  is  required  to  find  a  number  by  which  both  the  nume- 
rator and  denominator  may  be  divided.  Such  a  number  is 
called  a  common  divisor,  and  so  long  as  we  can  find  a  common 
divisor  to  the  numerator  and  the  denominator,  it  is  certain  that 
the  fraction  may  be  reduced  to  a  lower  form ;  but,  on  the  con- 


24  Algebra,  Sect.  1. 

trary,  when  we  see  that  except  unity  no  other  common  divisor 
can  be  found,  this  shews  that  the  fraction  is  already  in  the 
simplest  form  that  is  possible. 

91.  To  make  this  more  clear,  let  us  consider  the  fraction 
^Y^.  We  see  immediately  that  both  the  terms  are  divisible  by 
2,  and  that  there  results  the  fraction  |^.  Then  that  it  may 
again  be  divided  by  2,  and  reduced  to  ^|. ;  and  this  also,  having 
2  for  a  common  divisor,  it  is  evident,  may  be  reduced  to  -^-g. 
But  now  we  easily  perceive,  that  the  numerator  and  denomina- 
tor are  still  divisible  by  3  ',  performing  this  division,  therefore, 
we  obtain  the  fraction  |^,  which  is  equal  to  the  fraction  proposed^ 
and  gives  the  simplest  expression  to  which  it  can  be  reduced ; 
for  2  and  5  have  no  common  divisor  but  1,  which  cannot  dimin- 
ish these  numbers  any  farther. 

92.  This  property  of  fractions  preserving  an  invariable  value, 
whether  we  divide  or  multiply  the  numerator  and  denominator 
by  the  same  number,  is  of  the  greatest  importance,  and  is  the 
principal  foundation  of  the  doctrine  of  fractions.  For  example, 
we  can  scarcely  add  together  two  fractions,  or  subtract  them 
from  each  other,  before  we  have,  by  means  of  this  property, 
reduced  them  to  other  forms,  that  is  to  say,  to  expressions  whose 
denominators  are  equal.  Of  this  We  shall  treat  in  the  following 
chapter. 

93.  We  conclude  the  present  by  remarking,  that  all  integers 
may  also  be  represented  by  fractions.  For  example,  6  is  the 
same  as  4?  because  6  divided  by  1  makes  6  ;  and  we  may,  in  the 
same  manner,  express  the  number  6  by  the  fractions  y ,  >^,  ^-}, 
y ,  and  an  infinite  number  of  others  which  have  the  same  value. 


CHAPTER  IX. 

Of  the  Jlddition  and  Subtraction  of  Fractions. 

94.  When  fractions  have  equal  denominators,  there  is  no 
difficulty  in  adding  and  subtracting  them ;  for  ^  -f-  ^  is  equal  to 
4,  and  4  —  T  is  equal  to  f .     In  this  case,  either  for  addition  or 


I 


Chap.  9.  Of  Simple  ^luintitits,  25 

subtraction,  we  alter  only  the  numerators,  and  place  the  com- 
mon denominator  under  the  line ;  thus, 

T  Jit  +  ^U  —  T  A  —  tVV'  -f  T  A'  is  <^^l»al  to  ih  '  I J  —  tV  — 
n  +n  i«  equal  to  ||,  or  J|  ^  |J  —  A  —  iJ  +  i4  is  ^HiUal  to 
||,  or  I ;  also  4  + 1  is  equal  to  ■!•  or  1,  that  is  to  say,  an  inte- 
ger ;  and  |  —  |  -f- 1  is  equal  to  ^that  is  to  say,  nothing,  or  0. 

95.  But  when  fractions  have  not  *eqnal  denominator Sf  we  can 
always  change  tJiem  into  other  fractions  that  have  the  same  denomi- 
nator. For  example,  when  it  is  proposed  to  add  together  the 
fractions  ]  and  ^,  we  must  consider  that  |  is  the  same  as  |,  and 
that  I  is  equivalent  to  J  ;  we  have  therefore,  instead  of  the  two 
fractions  proposed,  these  |  -f-  f ,  the  sum  of  which  is  f .  If  the 
two  fractions  were  united  by  the  sign  minus,  as  |  —  4>  we 
should  have  |  —  f  or  |. 

Another  example  :  let  the  fractions  proposed  be  |  +  | ;  siiice 
J  is  tlie  same  as  |,  thig  value  may  be  substituted  for  it,  and  we 
may  say  |  +|  make  Y,  or  1  |. 

Suppose  farther,  that  the  sum  of  ^  and  i  were  required.  1 
say  that  it  is  /^  ;  for  4  makes  y\,  and  i  makes  ^.^. 

96.  TFe  may  have  a  greater  mimber  of  fractions  to  be  reduced  to  ft 
common  denominator ;  for  example,  |, -f,  |,  |,  |;  in  this  case 
the  whole  depends  on  finding  a  number  which  may  be  divisible  by 
all  the  denominators  of  those  fractions.  In  this  instance  60  is  the 
number  which  has  that  property,  and  which  consequently 
becomes  the  commpn  denominator.  We  shall  therefore  have 
|o  instead  of  J  ;  |§  instead  of  | ;  44  instead  of  | ;  *^  instead 
of  4  ;  and  ^  instead  of  f .  Tf  now  it  be  required  to  add  together 
all  these  fractions  |«,  ^J,  ||,  ||,  add  |  J  ;  we  have  only  to  add 
aU  the  numerators,  and  under  the  sum  place  the  common  deiwmi- 
nator  60;  that  is  to  say,  we  shall  have  V/,  or  three  integers 
and  1-3,  or  3  Aj. 

97.  The  whole  of  this  operation  consists,  as  we  before  stated, 
in  changing  two  fractions,  w  hose  denominators  are  unequal,  into 
tw^o  others,  whose  denominators  are  equal.  In  order  therefore 
to  perform  it  generally,  let  -^  and  -^  be  the  fractions  propos- 
ed. First,  multiply  the  two  terms  of  the  first  fraction  by  d,  we 
shall  have  the  fraction  ^^  equal  to  y  ;    next  multiply  tlie  two 

4  — 


26  Algebra.  Sect.  1. 

terms  of  the  second  fraction  by  h,  and  we  sliall  have  an  equiva- 
lent value  of  it  expressed  by  7^ ',  thus  the  two  denominators 
ai^e  become  equal.  Now,  if  the  sum  of  the  two  proposed  frac- 
tions be  required,  we  may  immediately  answer  that  it  is  — Ti—-  5 

bd 

and  if  their  difference  be  asked,  we  say  that  it  is  — "^ — .      If 

bd 

the  fractions  |  and  \,  for  example,  were  proposed,  we  should 
obtain  in  their  stead  ^f  and  4| ;  of  which  the  sum  is  ^V'*^*^^ 
the  difference  \\,  [8.] 

98.  To  this  part  of  the  subject  belongs  also  the  question,  of 
two  proposed  fractions,  which  is  the  greater  or  the  less ;  for,  to 
resolve  this,  we  have  only  to  reduce  the  two  fractions  to  the 
same  denominator.  Let  us  take,  for  example,  the  two  fractions 
I  arid  |. :  when  reduced  to  the  same  denominator,  the  first  be- 
comes |:J,  and  the  second  ^\,  and  it  is  evident  that  the  second, 
or  f,  is  the  greater,  and  exceeds  the  former  by  ■^-^, 

Again,  let  the  two  fractions  |  and  *  be  proposed.  We  shall 
have  to  substitute  for  them,  |J  and  || ;  whence  we  may  con- 
clude that  I  exceeds  |,  but  only  by  ^^, 

99.  When  it  is  required  to  subtract  a  fraction  from  an  integer, 
it  is  sufficient  to  change  one  of  the  unfts  of  that  integer  into  a  frac- 
tion which  has  the  same  denominator  as  that  which  is  to  be  sub- 
tracted ;  in  the  rest  of  the  operation  there  is  no  difficulty.  If  it 
be  required,  for  example,  to  subtract  |  from  1,  we  write  |  in- 
stead of  1,  and  say  that  |  taken  from  |  leaves  the  remainder  -J. 
So  -Z^,  subtracted  from  1,  leaves  -j^^. 

If  it  were  required  to  subtract  |  from  2,  we  should  write  1 
and  ^  instead  of  2,  and  we  should  immediately  see  that  after  the 
subtraction  there  must  remain  l\, 

100.  It  happens  also  sometimes,  that  having  added  two  or 
more  fractions  together,  we  obtain  more  than  an  integer ;  that 
is  to  say,  a  numerator  greater  than  tlie  denominator :  this  is  a 
case  which  has  already  occurred,  and  deserves  attention. 

We  found,  for  example,  article  96,  that  the  sum  of  the  five 
fractions  ^,  f ,  |»  y»  and  f  was  y/,  and  w^e  remarked  that  the 
value  of  this  sum  was  3  integers  and  ||,  or  JJ.  Likewise  |  + 
3,  or  -^^  4-  ^^2  wi^^cs  \l,  or  \^\,    We  have  only  to  perform  the 


Chap.  lOl  Of  Simple  ^antities,  27 

actual  division  of  the  numerator  by  the  denominator,  to  see  how 
many  integers  there  are  for  the  quotient,  and  to  set  down  the 
remainder.  Nearly  the  same  must  be  done  to  add  together 
numbers  compounded  of  integers  and  fractions ;  we  first  add 
the  fractions,  and  if  their  sum  produces  one  or  more  integers, 
these  are  added  to  the  other  integers.  Let  it  be  proposed,  for 
example,  to  add  3^  and  2| ;  we  first  take  the  §um  of  |  and  |, 
or  of  I  and  |.     It  is  J  or  1| ;  then  the  sum  total  is  6^. 


CHAPTER  X. 

Of  the  Multiplication  and  Division  of  Fractions, 

101.  The  rule  for  the  multiplication  of  a  fraction  by  an  integer  9 
or  whole  number,  is  to  multiphj  the  numerator  only  by  the  given 
number,  and  not  to  change  the  denominator :  thus, 

2  times,  or  twice  ^  makes  |,  or  1  integer  ; 

2  times,  or  twice  ^  makes  | ;  and 

3  times,  or  thrice  i  makes  |,  or  ^  ; 

4  times  j\  makes  4|  or  1-j?^,  or  1|. 

But,  instead  of  this  rule,  we  may  use  that  of  dividing  the  denom- 
inator by  the  given  integer  ;  and  this  is  preferable,  when  it  can  be 
used,  because  it  shortens  the  operation.  Let  it  be  required,  for 
example,  to  multiply  |-  by  3 ;  if  we  multiply  the  numerator  by 
the  given  integer  we  obtain  y',  which  product  we  must  reduce 
to  |.  But  if  we  do  not  change  the  numerator,  and  divide  the 
denominator  by  the  integer,  we  find  immediately  |,  or  2  |  for 
the  given  product.  Likewise  ^|  multiplied  by  6  gives  y ,  or  31. 

102.  In  general,  therefore,  the  product  of  the  multiplication 

of  a  fraction  -7  by  c  is  -7-;  and  it  may  be  remarked,  when  the 

integer  is  exactly  equal  to  the  denominator,  that  the  product  must 
he  equal  to  the  numerator, 

■]  I  taken  twice  gives  1 ; 
So  that  [- 1  taken  thrice  gives  2  ; 

J  1  taken  4  times  gives  3. 

And  in  general,  if  we  multiply  the  fraction  —  by  the  number 
h,  the  product  must  be  a,  as  we  have  already  shewn  5  for  since 


28  Algebra.  Sect.  1. 

—  expresses  the  quotient  resulting  from  the  division  of  the  divi- 
dend a  hy  the  divisor  b,  and  since  it  has  been  demonstrated  that 
the  quotient  multiplied  by  the  divisor  will  give  tlie  dividend,  it 

is  evident  that  ^  multiplied  by  b  must  produce  a. 

103.  We  have  shewn  how  a  fraction  is  to  be  multiplied  by  an 
integer ;  let  us  now  consider  also  how  a  fraction  is  to  he  divided 
by  an  integer  ;  this  inquiry  is  necessary  before  we  proceed  to  the 
multiplication  effractions  by  fractions.  It  is  evident,  if  I  have 
to  divide  the  fraction  |  by  2,  that  the  result  must  be  ^ ;  and 
that  the  quotient,  of  |.  divided  by  3  is  ^.  The  rule  therefore  is, 
to  divide  the  numerator  by  the  integer  without  changing  the  de- 
nominator.    Thus : 

41  divided  by  2  gives  -^y  ; 

II  divided  by  3  gives  ^\  ;  and 

II  divided  by  4  gives  -^j  ;  kc. 

104.  This  rule  may  be  easily  practised,  provided  the  nume- 
rator be  divisible  by  the  number  proposed  ;  but  very  often  it  is 
not :  it  must  therefore  be  observed  that  a  fraction  may  be  trans- 
formed into  an  infinite  number  of  other  expressions,  and  in  that 
number  there  must  be  some  by  which  the  numerator  might  be 
divided  by  the  given  integer.  If  it  were  required,  for  example, 
to  divide  |  by  2,  we  should  change  the  fraction  into  |,  and  then 
dividing  the  numerator  by  2,  we  should  immediately  have  4  for 
the  quotient  sought. 

In  general,  if  it  be  proposed  to  divide  the  fraction  -r  by  c,  we 

etc 
change  it  into  T->  and  then  dividing  the   numerator  ac  by  c, 

write  J-  for  the  quotient  sought. 

105.  When  therefore  a  fraction  -r  is  to  be  divided  by  an  integer 

c,  we  have  only  to  mnltiply  the  dercominator  by  that  number,  and 
leave  the  numerator  as  it  is.  Thus  |  divided  by  3  gives  3/^,  and 
|-  divided  by  5  gives  -/^. 

This  operation  becomes  easier  when  the  numerator  itself  is 
divisible  by  the  integer,  as  we  have  supposed  in  article  105. 


Chap.  10.  Of  Simple  ^antities.  29 

For  example,  -^^  divided  by  3  would  give,  according  to  our  last 
rule,  /^ ;  but  by  the  first  rule,  which  is  applicable  here,  we 
obtain  -j^^,  an  expression  equivalent  to  ^\,  but  more  simple. 

106.  AYe  shall  now  be  able  to  understand  how  one  fraction  -r 

0 

may  be  multiplied  by  another  fraction  -:.  We  have  only  to 
consider  that  -7  means  that  c  is  divided  by  d  ;  and  on  this  prin- 
ciple, we  shall  first  multiply  the  fraction  -r  by  c,  which  pro- 
duces  the  result  -r  ;  after  which  we  shall  divide  by  d,  which 

which  gives  r-^. 

Hence  the  following  rule  for  multiplying  fractions  ;  multiply 
separately  the  numerators  and  the  denominators. 
Thus  i  by  I  gives  the  product  f ,  or  4  ; 
I  by  4  makes  ^-^  ; 
I  by  ^%  produces  ||,  or  /^  ^  &c- 

107.  It  remains  to  shew  how  owe  fraction  may  be  divided  by 
another.  We  remark  first,  that  if  the  two  fractious  have  the  same 
number  for  a  denominator,  the  division  takes  place  only  with 
respect  to  the  numerators  ;  for  it  is  evident,  that  ^^^  are  contain- 
ed as  many  times  in  -^^  as  3  in  9,  that  is  to  say,  thrice ;  and,  in 
the  same  manner,  in  order  to  divide  -^^  by  ^^,  we  have  only  to 
divide  8  by  9,  which  gives  |.  We  shall  also  have  /^  in  U,  3 
times  :  ^  J^  in  ^%%,  7  times  ;  -^j  in  ^%,  f  ;  &c. 

108.  But  when  the  fractions  have  not  equal  denominators^  we 
must  have  recourse  to  the  method  already  mentioned  for  reduc- 

inj^  them  to  a  common  denominator.     Let  there  be,  for  exam- 

a  c 

pie,  the  fraction  -r  to  be  divided  by  the  fraction  -z ;  we  first  re- 
duce them  to  the  same  denominator ;  we  have  then  r-j  to  be 

cb 
divided  by  ir;  it  is  now  evident,  that  the  quotient  must  be 

ad 
represented  simply  by  the  division  of  ad  by  be  ;  which  gives  r-. 


30  Ss^ebra,  Sect.  1. 


"£> 


Hence  the  following  rule  :  Multiply  the  numerator  of  the  divi- 
dend by  the  denominator  of  the  divisor,  and  the  denominator  of  the 
dividend  by  the  nnmerator  of  the  divisor  ;  the  first  product  will  be 
the  numerator  of  the  quotient,  and  the  second  will  be  its  denomi- 
nator, 

109.  Applying  this  rule  ts  the  division  of  4  by  |,  we  shall 
have  the  quotient  ^|^ ;  the  division  of  |  by  ^  will  give  f  or  |  or 
1  and  I ;  and  i|  by  f  will  give  »l«,  or  |. 

110.  This  rule  for  division  is  often  represented  in  a  manner 
more  easily  remembered,  as  follows  :  Invert  the  fraction  which 
is  the  divisor,  so  that  the  denominator  may  be  in  the  place  of  the 
numerator,  and  the  latter  be  written  under  the  line  ;  then  multiply 
the  fraction,  which  is  the  dividend  by  this  inverted  fraction,  and 
the  product  will  be  the  quotient  sought.  Thus  |  divided  by  J  is 
the  same  as  |  multiplied  by  ^,  which  makes  |,  or  1  |.  Also  4 
divided  by  |  is  the  same  as  |  multiplied  by  |,  which  is  ^f  ;  or 
51  divided  by  f  gives  the  same  ||  multiplied  by  |,  the  product 
of  which  is  ||§,  or  |. 

We  see  then,  in  general,  that  to  divide  by  the  fraction  -|,  is  the 
same  as  to  multiply  by  ^,  or  2  ;  that  division  by  -i  amounts  to  mul- 
iiplication  by  4,  or  by  3,  <^c, 

111.  The  number  100  divided  by  |  will  give  200;  and  1000 
divided  ^  will  give  3000.  FuHher,  if  it  were  required  to  divide 
1  by  -i/^^,  the  quotient  would  be  1000;  and  dividing  1  by 
T^W^TT'  t^®  quotient  is  100000.  This  enables  us  to  conceive 
that,  when  any  number  is  divided  by  0,  the  result  must  be  a 
number  infinitely  great ;  for  even  the  division  of  1  by  the  small 
fraction  ^^--^^^^-^  gives  for  the  quotient  the  very  great  num- 
ber 1000000000. 

112.  Every  number  when  divided  by  itself  producing  unity, 
it  is  evident  that  a  fraction  divided  by  itself  must  also  give  1  for 
the  quotient.  The  same  follows  from  our  rule  :  for,  in  order  to 
divide  |  by  |,  we  must  multiply  |  by  4,  and  we  obtain  if,  or  1  ; 

and  if  it  be  required  to  divide  -j  by  "r>  we  multiply  -r  by  — ; 
now  the  product    .  is  equal  to  1. 


Chap.  11. 


Of  Simple  ^uatitities. 


31 


113.  We  have  still  to  explain  an  expression  which  is  fre- 
quently used.  It  may  be  asked,  for  example,  what  is  the  half 
of  I ;  this  means  that  we  must  multiply  |  by  ^.  So  likewise,  if 
the  value  of  |  of  |  were  required,  we  should  multiply  |  by  |, 
which  produces  JJ  ;  and  |  of  ^^  is  the  same  as  /^  multiplied  by 
|,  which  produces  fj. 

114.  Lastly,  we  must  here  observe,  with  respect  to  the  signs 
-f  and  — ,  tlie  same  rules  that  we  before  laid  down  for  integers. 
Thus  +1  multiplied  by  —  4,  makes  —  |  ;  and  —  |  multiplied  by 
—  ^,  gives  4-  ^%.  Farther,  —  |  divided  by  +  |,  makes  —  4f  ; 
and  —  I  divided  by  —  |,  makes  +  -Jf  or  +  1. 


CHAPTER  XI. 

Of  Square  JVhimbers, 

115.  The  product  of  a  number^  when  multiplied  by  itself,  is 
called  a  square  ;  and  for  this  reason,  the  number,  considered  in 
relation  to  such  a  product,  is  called  a  square  root. 

For  example,  when  we  multiply  12  by  12,  the  product  144  is 
a  square,  of  which  the  root  is  12. 

This  term  is  derived  from  geometry,  which  teaches  us,  that 
the  contents  of  a  square  arc  found  by  multiplying  its  side  by 
itself. 

116.  Square  numbers  are  found  therefore  by  multiplication  ; 
that  is  to  say,  by  multiplying  the  root  by  itself.  Thus  1  is  the 
square  of  1,  since  1  multiplied  by  1  makes  1 ;  likewise,  4  is  the 
square  of  2  ;  and  9  the  square  of  3  ;  2  also  is  the  root  of  4,  and 
3  is  the  root  of  9. 

We  shall  begin  by  considering  the  squares  of  natural  numbers, 
and  shall  first  give  the  following  small  table,  on  the  first  line  of 
wiiich  several  numbers,  or  roots,  are  ranged,  and  on  the  second 
their  squares.[9.] 


Numbers. 
Squares. 

1 

1 

2 
4 

3 
9 

4 
16 

5 
25 

6 
36 

7 
49 

8  9  10 
64  81  100 

11 
121 

12 
144 

13 

169 

32 


Ms^ebra, 


Sect  1. 


lir.  It  will  be  readily  perceived,  that  the  series  of  square 
numbers  thus  arranged  has  a  singular  property ;  namely,  that 
if  each  of  them  be  subtracted  from  that  which  immediately 
follows,  the  remainders  always  increase  by  2,  and  form  this 
series ; 

3,  5,  7,  9,  11,  13,  15, 17,  19,  21,  &c. 
118.  The  squares  of  fractions  are  found  in  the  same  manner,  by 
multiplying  any  given  fraction  by  itself   For  example,  the  square 
isl. 


ofi 


The  square  of 


1  h^ 

1} 

4,     -' 


1    . 

r  > 


4   . 
•5-  ? 


1     . 

IT  ' 


We  have  only  therefore  to  divide  the  square  of  the  numerator 
by  the  square  of  the  denominator,  and  the  fraction,  which  ex- 
presses that  division,  must  be  the  square  of  the  given  fraction. 
Thusj  II  is  the  square  of  | ;  and  reciprocally,  |  is  the  root 
nf  2* 

119.  When  the  square  of  a  mixt  number,  or  a  number,  com- 
posed of  an  integer  and  a  fraction,  is  required,  we  have  only  to 
reduce  it  to  a  single  fraction,  and  then  take  the  square  of  that 
fraction.  Let  it  be  required,  for  example,  to  find  the  square  of 
2|  ;  we  first  express  this  number  by  |,  and  taking  the  square 
of  that  fraction,  we  have  y,  or  6  A,  for  the  value  of  the  square 
of  2J.  So  to  obtain  the  square  of  31,  we  say  3^  is  equal  to  y  ; 
therefore  its  square  is  equal  to  y/,  or  to  10  and  -f^.  The 
'  squares  of  the  numbers  between  3  and  4,  supposing  them  to 
increase  by  one  fourth,  are  as  follows  : 


Numbers. 
Squares. 

3 

H 

H 

3| 

4 

9 

'"tV 

m 

144t 

16 

From  this  small  table  we  may  infer,  that  if  a  root  contain  a 
fraction,  its  square  also  contains  one.  Let  the  root,  for  example, 
be  1^*2  ?  its  square  is  4||,  or  2yJ^ ;  that  is  to  say,  a  little 
greater  than  the  integer  2. 

120.  Let  us  proceed 'to  general  expressions.  When  the  root 
is  a,  the  square  must  be  aa  ;  if  the  root  be  2a,  the  square  is  4aa ; 


Chap.  12.  Of  Simple  (Quantities.  33 

which  shews  that  by  doubling  the  root,  the  square  becomes  4 
times  greater.  So  if  the  root  be  Scr,  the  square  is  9aa  ;  and  if 
the  root  be  4a,  the  square  is  I6aa.  But  if  the  root  be  a6,  the 
square  is  aahb  ;  and  if  the  root  be  ahc^  the  square  is  aahhcc. 

121.  Thus  when  the  root  is  composed  of  two,  or  more  factors, 
we  multiply  their  squares  together  ;  and  reciprocally,  if  a  square 
be  composed  of  two,  or  more  factors,  of  which  each  is  a  square,  we 
have  only  to  mvlliply  together  the  roots  of  those  squares,  to  obtain 
the  complete  root  of  the  square  proposed.  Thus,  as  2304  is  equal 
to  4  X  16  X  36,  the  square  root  of  it  is  2  x  4x6,  or  48  j  and 
48  is  found  to  be  the  true  square  root  of  2304,  because  48  x  48 
gives  2304. 

122.  Let  us  now  consider  what  must  be  observed  on  this  sub- 
ject with  regard  to  the  signs  +  and  — .  First,  it  is  evident 
that  if  the  root  has  tlie  sign  -{-,  that  is  to  say,  is  a  positive  num- 
ber, its  square  must  necessarily  be  a  positive  number  also, 
because  +  by  -f-  makes  + :  the  square  of  +  a  will  be  +  aa. 
But  if  the  root  be  a  negative  number,  as  —  a,  the  square  is  still 
positive,  for  it  is  +  aa  ;  we  may  therefore  conclude,  that  -f  aa 
is  the  square  both  q/*  -f-  a  and  of  —  a,  and  that  consequently  every 
square  has  two  roots,  one  positive  and  the  other  negative.  The 
square  root  of  25,  for  example,  is  both  +  5  and  —  5,  because 
—  5  multiplied  by  —  5  gives  25,  as  well  as  +  5  by  -f  5. 


CHAPTER  XII. 

Of  Square  Roots,  and  of  Irrational  J^umbers  resulting  from  them, 

123.  What  we  have  said  in  tlie  preceding  chapter  is  chiefly 
this  :  that  the  square  root  of  a  given  number  is  nothing  but  a 
number  whose  square  is  equal  to  the  given  number ;  and  that 
we  may  put  before  those  roots  either  the  positive  or  the  negative 
sign. 

124.  So  that  when  a  square  number  is  given,  provided  we 
retain  in  our  memory  a  sufficient  number  of  square  numbers,  it 
is  easy  to  find  its  root.  If  106,  for  example,  be  the  given  num- 
ber, we  know  that  its  square  root  is  14. 

5 


54  Mgehru,  Sect.  1. 

Fractions  likewise  are  easily  managed :  it  is  evident,  for 
example,  that  ^  is  the  square  root  of  f  f .  To  he  convinced  of 
this,  we  have  only  to  take  the  square  root  of  the  numerator,  and 
that  of  the  denominator. 

If  the  number  proposed  be  a  mixt  number,  as  121,  we  reduce 
it  to  a  single  fraction,  which  here  is  Y>  and  we  immediately 
perceive  that  |,  or  3|,  must  be  the  square  root  of  12|. 

125.  But  when  the  given  number  is  not  a  square,  as  12  for 
example,  it  is  not  possible  to  extract  its  square  root ;  or  to  find 
a  number,  which,  multiplied  by  itself,  will  give  the  product  12. 
We  know,  however,  that  the  square  root  of  12  must  be  greater 
than  3,  because  3x3  produces  only  9  j  and  less  than  4,  because 
4x4  produces  16,  which  is  more  than  12.  We  know  also,  that 
this  root  is  less  than  3| ;  for  we  have  seen  that  the  square  of 
S|,  or  J  is  12A.  Lastly,  we  may  approach  still  nearer  to  this 
root,  by  comparing  it  with  3yL  ;  for  the  square  of  3-/^,  or  of  4| 
is  YfV'  OJ'  12  ^l-g,  so  that  this  fraction  is  still  greater  than  the 
root  required  ;  but  very  little  greater,  as  the  difference  of  the 
two  squares  is  only  ■^■■^j, 

126.  We  may  suppose  that  as  3  J  and  3^^^  are  numbers  greater 
than  the  root  of  12,  it  might  be  possible  to  add  to  3  a  fraction  a 
little  less  than  -^j,  and  precisely  such,  that  the  square  of  the 
sum  would  be  equal  to  12. 

Let  us  therefore  try  with  S^,  since  4  is  a  little  less  than  ^j. 
Now  3|  is  equal  to  y,  the  square  of  which  is  *^\^ ,  and  conse- 
quently less  by  ||  than  12,  which  may  be  expressed  by  ^^y. 
It  is  therefore  proved  that  3^  is  less,  and  that  3/^  is  greater 
than  the  root  required.  Let  us  then  try  a  number  a  little  greater 
than  3^)  but  yet  less  than  5^^,  for  example,  3-^^,  This  number, 
which  is  equal  to  41,  has  for  its  square  V/t'.  Now,  by  reduc- 
ing 12  to  this  denominator,  we  obtain  W^^  ;  which  shews  that 
3^*y  is  still  less  than  the  root  of  12.  viz.  by  y|y.  Let  us  there- 
fore substitute  for  ^\  the  fraction  -f^,  which  is  a  little  greater, 
and  see  what  will  be  the  result  of  the  comparison  of  the  square  of 
5-^«y  with  the  proposed  number  12.  The  square  of  3^\  is  Y//  ; 
now  12  reduced  to  the  same  denominator  is  ^y  ;  so  that  3-/^  is 
still  too  small,  thougli  only  by  y|^,  whilst  S^-g  has  been  found 
too  great. 


Chap.  12.  Of  Simple  Quantities,  35 

127.  It  is  evident  therefore,  that  whatever  fraction  be  joined 
to  3,  the  square  of  tliat  sum  must  always  contain  a  fraction,  and 
can  never  be  exactly  equal  to  the  integer  12.  Thus,  although 
we  know  tliat  the  square  root  of  12  is  greater  than  5-fj  and  less 
than  3^\,  yet  we  are  unable  to  assign  an  intermediate  fraction 
between  these  two,  which,  at  the  same  time,  if  added  to  3,  would 
express  exactly  the  square  root  of  12.  Notwithstanding  this, 
we  are  not  to  assert  that  the  square  root  of  12  is  absolutely  and 
in  itself  indeterminate ;  it  only  follows  from  what  has  been  said, 
that  this  root,  though  it  necessarily  has  a  determinate  magni- 
tude, cannot  be  expressed  by  fractions. 

128.  There  is  therefore  a  sort  of  numbers  wkich  cannot  he 
assigned  by  fractions,  and  which  are  nevertheless  determinate 
quantities;  the  square  root  of  12  furnishes  an  example.  We 
call  this  new  species  of  numbers,  irrational  numbers ;  they  occur 
whenever  we  endeavour  to  find  the  square  root  of  a  number 
which  is  not  a  sc^uare.  Thus,  2  not  being  a  perfect  square,  the 
square  root  of  2,  or  the  number  which,  multiplied  by  itself, 
would  produce  2,  is  an  irrational  quantity.  These  numbers  are 
also  called  surd  quantities,  or  incommensurables, 

129.  These  irrational  quantities,  though  they  cannot  be  ex« 
pressed  by  fractions,  are  nevertheless  magnitudes,  of  which  we 
may  form  an  accurate  idea.  For  however  concealed  the  square 
root  of  12,  for  example,  may  appear,  we  are  not  ignorant,  that  it 
must  be  a  number  which,  when  multiplied  by  itself,  would 
exactly  produce  12  ;  and  this  property  is  sufficient  to  give  us  an 
idea  of  the  number,  since  it  is  in  our  power  to  approximate  its 
value  continually. 

130.  As  we  are  therefore  sufficiently  acquainted  with  the  nature 
of  the  irrational  numbers,  under  our  present  consideration,  a  par- 
ticular sign  has  been  agreed  on,  to  express  the  square  roots  of  all 
numbers  that  are  not  perfect  squares.  This  sign  is  written 
thus  v>  and  is  read  square  root.  Thus,  ^12  represents  the 
square  root  of  12,  or  the  number  which,  multiplied  by  itself, 
produces  12.  So,  ^^  represents  the  square  root  of  2 ;  ^3"  that 
of  3  ;  ^/|  that  of  f  ;  and,  in  general,  ^T  represents  the  square 
root  of  the  mmber  a.     Whenever  therefore  we  would  express  the 


.-»% 


36  Algebra,  Sect.  l.i 

square  root  of  a  number  which  is  not  a  square,  we  need  only 
make  use  of  the  mark  v^  hy  jilacing  it  before  the  number. 

131.  The  explanation,  which  we  have  given  of  irrational  num- 
bers, will  readily  enable  us  to  apply  to  them  the  known  methods 
of  calculation.  For  knowing  that  the  square  root  of  2,  multi- 
plied by  itself,  must  produce  2 ;  we  know  also,  that  the  multipli- 
cation y^Y  by  x/T  must  necessarily  produce  2  ;  that,  in  the  same 
manner,  the  multiplication  \/T  by  y/T  must  give  3 ;  that  \/T  by 
\/T  makes  5  ;  that  v|  ^7  Vi  makes  | ;  and,  in  general,  that 
\/T  multiplied  by  ^T  produces  a. 

132.  But  when  it  is  required  to  multiply  s/T  by  vtT  the  product 
will  be  found  to  be  y/jTb ;  because  we  have  shewn  before,  that  if  a 
square  has  two  or  more  factors,  its  root  must  be  composed  of 
the  roots  of  those  factors.  Wherefore  we  find  the  square  root 
of  the  product  ab,  which  is  v^^?  hy  multiplying  the  square  root 
of  a  or  v^^  by  the  square  root  of  b  or  \/T  It  is  evident  from 
this,  that  if  b  were  equal  to  a,  we  should  have  \/aa  for  the  pro- 
duct of  \/^  by  vZT  Now  v^  is  evidently  a,  since  aa  is  the 
square  of  a. 

133.  In  division,  if  it  were  required  to  divide  \/^  for  exam- 
ple, by  V67  we  obtain  ^±;  and  in  this  instance  the  irration- 
ality may  vanish  in  the  quotient.  Thus,  having  to  divide  x/Ts 
by  y/sT  the  quotient  is  vV'  which  is  reduced  to  v/|>  and  conse- 
quently to  |,  because  |  is  the  square  of  |. 

134.  When  the  number,  before  which  we  have  placed  the 
radical  sign  >/,  is  itself  a  square,  its  root  is  expressed  in  the  usual 
way.  Thus  \/T  is  the  same  as  2 ;  x/T  the  same  as  3 ;  V36  the 
same  as  6  ;  and  vi2-J-  the  same  as  J,  or  3J.  In  these  instances 
the  irrationality  is  only  apparent,  and  vanishes  of  course. 

135.  It  is  easy  also  to  multiply  irrational  numbers  by  ordi- 
nary numbers.  For  example,  2  multiplied  by  \/T  makes  2  x/T, 
and  3  times  ^2*  makes  3  vsT  In  the  second  example,  however, 
as  3  is  equal  to  x^g^  we  may  also  express  3  times  vF  by  x^W 
multiplied  by  vsT  or  by  v/i8.  So  2  x^7  is  the  same  as  x/Ta,  and 
3  x/^  the  same  as  V9^.  And,  in  general,  b  v^  has  the  same 
'value  as  the  square  root  of  bba,  or  v^bT;  whence  we  infer  recip- 
rocally, that  when  the  number  which  is  preceded  by  the  radical 


Chap.  12. 


Of  Simple  Quantities, 


57 


sign  contains  a  square,  we  may  take  the  root  of  that  square  and 
put  it  before  the  sign,  as  we  should  do  in  writing  b  x/a'  instead 
of  v^^oT  After  this,  the  following  reductions  will  be  easily 
understood  : 


vs; 

or  ^2-4 

2  V2; 

\/12", 

or  ^3-4 

2  \/3"; 

\/24, 

or   V2-9 
or  V6-4 

>  is  equal  to  < 

3  V2"; 
2  \/6"; 

V32, 

or  \/2-l6 

4  V2; 

V75, 

or   V3-23  ^ 

^  5  VS"; 

and  so  on. 

136.  Division  is  founded  on  the  same  principles.    \/ 3,  divided 


by  Vb,  ^^^'es  -|=,  or 

Vb 


\/8        1 


Farther 


VI8 

V/12 

V3" 

2 

V2" 

3 

V3" 


\^.    In  the  same  manner. 


4j|»orV4,  or  2; 


J>  is  equal  to  <(   J-,  or  V9,  or  3  ; 

J  12 
"3^ 


>>  is  equal  to  <{   — =>  or  v|j  or  v/3 

V3 


or  V4,  or  2. 


V/4  A  — 

:r^,  or  v|,  or  V2  ,• 

V9' 


^^,  or  v*$*5  or  V24, 


12 

or  \/&4,    or  lastly  2  ve" . 

137.  There  is  nothing  in  particular  to  be  observed  with 
respect  to  the  addition  and  subtraction  of  such  quantities,  be- 
cause we  only  connect  them  by  the  signs  +  and  — ^.  For 
example,  v/2"  added  to  ^/J"  is  written  ^/T  +  \/T  ;  and  x/T  sub' 
tracted  from  \/T  is  xvritten  \/T  —  V37 


38  Mgehra,  Sect.  1. 

138.  We  may  observe  lastly,  that  in  order  to  distin.i^uish  the 
irrational  numbers,  we  call  all  other  numbers,  both  integral  and 
fractional,  rational  numbers. 

So  that,  whenever  \^e  speak  of  rational  numbers,  we  under- 
stand integers  or  fractions. 


CHAPTER.  XIII. 

Of  Impossible  or  Imaginary  Quantities,  which  arise  from  the 

same  source. 

139.  We  have  already  seen  that  the  squares  of  numbers, 
negative  as  well  as  positive,  are  always  positive,  or  affected 
with  the  sign  + ;  having  shewn  that  —  a  multiplied  by  —  a 
gives  -f  aa,  the  same  as  the  product  of  -f-  a  by  +  a.  Wherefore, 
in  the  preceding  chapter,  we  supposed  that  all  the  numbers,  of 
which  it  was  required  to  extract  the  square  roots,  were  positive. 

140.  When  it  is  required  therefore  to  extract  the  root  of  a 
negative  number,  a  very  great  difficulty  arises ;  since  there  is 
no  assignable  number,  the  square  of  which  would  be  a  negative 
quantity.  Suppose,  for  example,  that  we  wished  to  extract  the 
root  of  —  4  5  we  require  such  a  number,  as  when  multiplied  by 
itself,  would  produce  — 4  ;  now  this  number  is  neither  -f  2  nor 
—  2,  because  the  square,  both  of  +  2  and  of  — -2,  is  -f-  4,  and 
not  —  4. 

141.  We  must  therefore  conclude,  that  the  square  root  of  a 
iiegative  number  cannot  be  either  a  positive  number,  or  a  negative 
number f  since  the  squares  of  negative  immbers  also  take  the  sign 
plus.  Consequently  the  root  in  question  must  belong  to  an  entirely 
distinct  species  of  numbers ;  since  it  cannot  be  ranked  either 
among  positive,  or  among  negative  numbers. 

142.  Now,  we  before  remarked,  that  positive  numbers  are  all 
greater  than  nothing,  or  0,  and  that  negative  numbers  are  all  less 
than  nothing,  or  0  ;  so  that  whatever  exceeds  0,  is  expressed  by 
positive  numbers,  and  whatever  is  less  than  0,  is  expressed  by 
negative  numbers.  The  square  roots  of  negative  numbers, 
therefore,  are  neither  greater  nor  less  than  nothing.     We  can- 


Chap.  15.  Of  Simple  Quantities,  39 

not  say  however,  that  they  are  0 ;  for  0  multiplied  hy  0  pro- 
duces 0,  and  consequently  does  not  give  a  negative  number. 

143.  Now,  since  all  numbers,  which  it  is  possible  to  conceive, 
are  either  greater  or  less  than  0,  or  are  0  itself,  it  is  evident 
that  we  cannot  rank  the  square  root  of  a  negative  number 
amongst  possible  numbers,  and  we  must  therefore  say  that  it  is 
an  impossible  quantity.  In  this  manner  we  are  led  to  the  idea 
of  numbers  which  from  their  nature  are  impossible.  These  num- 
bers are  usually  called  imaginartj  quantitieSf  because  they  exist 
merely  in  the  imagination. 

144.  All  such  expressions,  as  v/~l,  V— 2,  V— 3,  V— 4,  &c. 
are  consequently  impossible,  or  imaginary  numbers,  since  they 
represent  roots  of  negative  quantities  :  and  of  such  numbers  we 
may  truly  assert,  that  they  are  neither  nothing,  nor  greater  tl»an 
nothing,  nor  less  than  nothing ;  which  necessarily  constitutes 
them  imaginary,  or  impossible. 

145.  But  notwithstanding  all  this,  these  numbers  present 
themselves  to  the  mind ;  they  exist  in  our  imagination,  and  we 
still  have  a  sufficient  idea  of  them  ;  since  we  know  that  by  \/—4, 
is  meant  a  number  which,  multiplied  by  itself,  produces  —  4. 
For  this  reason  also,  nothing  prevents  us  from  making  use  of 
these  imaginary  numbers,  and  employing  them  in  calculation. 

146.  The  first  idea  that  occurs  on  the  present  subject  is,  that 
the  square  of  V— 3,  for  example,  or  the  product  of  v/~3  hy 
V— 3,  must  be  —  3 ;  that  the  product  of  vZi  by  \/~i  is  — 1  ; 
and,  in  general,  that  by  multiplying  v— a  by  \/—a,  or  by  taking 
the  square  of  \/^a,  we  obtain  —  a. 

147.  Now,  as  —  a  is  equal  to  -f  a  multiplied  by  —  1,  and  as 
the  square  root  of  a  product  is  found  by  multiplying  together 
the  roots  of  its  factors,  it  follows  that  the  root  of  a  mul- 
tiplied by  —  1,  or  \/— a,  is  equal  to  ^'^  multiplied  by  yZ—l. 
Now  v/o" is  a  possible  or  real  number,  consequently  the  whole 
impossibility  of  an  imaginary  quantity  may  be  always  reduced  to 
v/— 1.  For  this  reason,  \/— 4  is  equal  to  \/4~  multiplied  by 
V— 1,  and  equal  to  2  V—l,  o»  account  of  ^^  being  equal  to  2, 
For  the  same  reason,  ^—g  is  reduced  to  ^g'  x  V^  ^^  ^ 
\/— .1 ;  and  v—is  is  equal  to  4  \/Zi. 


40  Algch^a,  Sect.  1. 


-o 


148.  Moreover,  as  \/'^  multiplied  by  \/T  makes  v/o*?  we 
shall  have  v/6~  for  value  of  v— 2  multiplied  by  yCIs  ;  and  ^47  or 
2,  for  the  value  of  the  product  of  '^—1  by  V--4'  We  see,  there- 
fore^ that  two  imaginary  numbers,  multiplied  together,  produce  a 
real,  or  possible  one. 

But,  on  the  contrary,  a  possible  number,  multiplied  by  an  im- 
possible number,  gives  always  an  imaginary  product :  thus,  V—S 
by  X/+5  gives  \/1^. 

149.  It  is  the  same  with  regard  to  division ;    for  v'«~divided 

by  x/r  making  p,  it  is  evident  that  v^  divided  v~i  will 
make  \/+4,  or  2  5  that  v+3  divided  by  >/Il3  will  give  \/.^ 
and  that  1  divided  by  v/— l  gives     f"^^,  or  v—l ;  because  1  is 

equal  to  v+i* 

150.  We  have  before  observed,  that  the  square  root  of  any 
number  has  always  two  values,  one  positive  and  the  other 
negative  ;  that  \/4i  for  example,  is  both  +  2  and  —  2,  and  that, 
in  general,  we  must  take  —  Vo"  as  well  as  +  \/^  for  the  square 
root  of  a.  This  remark  applies  also  to  imaginary  numbers ; 
the  square  root  of —  a  is  both  +  v—a  and  < —  \/— a;  but  we  must 
not  confound  the  signs  -f  and  — ,  which  are  before  the  radical  sign 
x/,  with  the  sign  which  comes  after  it, 

151.  It  remains  for  us  to  remove  any  doubt  which  may  be 
entertained  concerning  the  utility  of  the  numbers  of  which  we 
have  been  speaking;  for  those  numbers  being  impossible,  it 
would  not  be  surprising  if  any  one  should  think  them  entirely 
useless,  and  the  subject  only  of  idle  speculation.  This  however  is 
not  the  case.  The  calculation  of  imaginary  quantities  is  of  the 
greatest  importance :  questions  frequently  arise,  of  which  we 
cannot  immediately  say,  whether  they  include  any  thing  real 
and  possible,  nor  not.  Now,  when  the  solution  of  such  a  ques- 
tion leads  to  imaginary  numbers,  we  are  certain  that  what  is 
required  is  impossible. 

In  order  to  illustrate  wliat  we  have  said  by  an  example,  sup- 
pose it  were  proposed,  to  divide  t]ie  number  12  into  two  such 
parts,  that  tlie  product  of  those  parts  may  be  40.  If  we  resolve 
this  question  by  the  ordinary  rules,  we  find  for  the  parts  sought 


Chap.  14. 


Of  Simple  Quantities. 


41 


6  +  v/~4  and  6  — \/--4;  but  these  numbers  are  imaginary  :  we 
conclude  therefore  that  it  is  impossible  to  resolve  the  question. 
The  difference  will  be  easily  perceived,  if  we  suppose  the 
question  had  been  to  divide  12  into  two  parts  which,  multiplied 
together,  would  produce  35  :  for  it  is  evident  that  those  parts 
must  be  7  and  5. 


CHAPTER  XIV. 

Of  Cubic  JSTumhers, 

152.  "When  a  number  has  been  multiplied  thrice  by  itself 9  or, 
which  is  the  same  thing,  when  the  square  of  a  number  has  been^ 
multiplied  once  more  by  that  number,  we  obtain  a  product  which  is 
called  a  cube,  or  a  cubic  number.  Thus,  the  cube  of  a  is  aaa,  since 
it  is  the  product  obtained  by  multiplying  a  by  itself,  or  by  a,  and 
that  square  aa  again  by  a. 

The  cubes  of  the  natural  numbers  therefore  succeed  each 
other  in  the  following  order. 


Numbers. 
Cubes. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

1 

8 

27 

64 

125 

216 

343 

512 

729 

1000 

153.  If  we  consider  the  differences  of  those  cubes,  as  we  did 
of  the  squares,  by  subtracting  each  cube  from  that  which  come& 
after  it,  we  obtain  the  following  series  of  numbers  : 

7,  19,  37,  61,  91,  127,  169,  217,  271. 
At  first,  we  do  not  observe  any  regularity  in  them  ;  but  if  we 
take  the  respective  differences  of  these  numbers,  we  find  the 
following  series : 

12,  18,  24,  30,  36, 42, 48,  54,  60  ; 
in  which  the  terms,  it  is  evident,  increase  always  by  6. 

154.  After  the  definition  we  have  given  of  a  cube,  it  will  not 
be  difficult  to  find  the  cubes  of  fractional  numbers  ;  ^  is  the  cube 
of  I ;  ^V  ^^  ^^®  ^^^®  ^^  i  ?  ^^d  ^\  is  the  cube  of  |.  In  the 
same  manner,  we  have  only  to  take  the  cube  of  the  numerator 
and  that  of  the  denominator  separately,  and  we  shall  have  |J 
for  the  cube  of  |. 

6 


42  Algebra.  Sect.  1, 

155.  If  it  be  required  tojind  the  cube  of  a  miootnumberf  we  must 
first  reduce  it  to  a  single  fraction,  and  then  proceed  in  the  manner 

thai  has  been  described.  To  find,  for  example,  the  cube  of  1^, 
we  must  take  that  of  |,  which  is  y ,  or  3  and  |.  So  the  cube 
of  1^,  or  of  the  single  fraction  ^,  is  y/,  or  1|| ;  and  the  cube 
of  31,  or  of  y  is  « J|^  or  34fl. 

156.  Since  aaa  is  the  cube  of  a,  that  of  ab  will  be  aaabbb ; 
whence  we  see,  that  if  a  number  has  two  or  more  factors,  we 
may  find  its  cube  by  multiplying  together  the  cubes  of  those  factors. 
For  example,  as  12  is  equal  to  3  x  4,  we  multiply  the  cube  of  3, 
which  is  27,  by  the  cube  of  4,  which  is  64,  and  we  obtain  1728, 
the  cube  of  12.  Further,  the  cube  of  2a  is  Saaa,  and  conse- 
quently 8  times  greater  than  the  cube  of  a ;  and  likewise,  the 
cube  of  Sa  is  27aaa,  that  is  to  say,  27  times  greater  than  the 
cube  of  a, 

157.  Let  us  attend  here  also  to  the  signs  -f-  and  — .  It  is 
evident  that  the  cube  of  a  positive  number  +  a  must  also  be 
positive,  that  is  +  aaa.  But  if  it  be  required  to  cube  a  negative 
number  —  a,  it  is  found  by  first  taking  the  square,  which  is 
-I-  aa,  and  then  multiplying,  according  to  the  rule,  this  square 
by  —  a,  which  gives  for  the  cube  required  —  aaa.  In  this 
respect,  therefore,  it  is  not  the  same  with  cubic  numbers  as  with 
squares,  since  the  latter  are  always  positive :  whereas  the  cube  of 
—  1  is  —  1,  that  of  —  2  is  —  8,  that  of —  3  is  —  27,  and  so  on. 


CHAPTER  XV. 

Of  Cube  Boots,  and  of  Irrational  Mimbers  resulting  from  them. 

158,  As  we  can,  in  the  manner  already  explained,  find  the 
cube  of  a  given  number,  so,  when  a  number  is  proposed,  we  may 
also  reciprocally  find  a  number,  which,  multiplied  thrice  by  itself, 
will  produce  that  number.  The  number  here  sought  is  called, 
with  relation  to  the  other,  the  cube  root.  So  that  the  cube  root  of 
a  given  number  is  the  number  whose  cube  is  equal  to  that  given 
number. 


Chap.  15.  Of  Simjyle  Quantities,  4S 

159.  It  is  easy  therefore  to  determine  the  cube  root,  when  the 
number  proposed  is  a  real  cube,  such  as  the  examples  in  the  last 
chapter.  For  we  easily  perceive  that  the  cube  root  of  1  is  1 ; 
that  of  8  is  2  ;  that  of  27  is  3  ;  that  of  64  is  4,  and  so  on.  And, 
in  the  same  manner,  the  cube  root  of —  27  is  —  3  ;  and  that  of 
—  125  is  —  5. 

Farther,  if  the  proposed  number  be  a  fraction,  as  ^»y,  the  cube 
root  of  it  must  be  | ;  and  that  of  j^-^\  is  ^,  Lastly,  the  cube 
root  of  a  mixt  number  2|^  must  be  |,  or  1^  ;  because  2|^  is 
equal  to  -|^. 

160.  But  if  the  proposed  number  be  not  a  cube,  its  cube  root 
cannot  be  expressed  either  in  integers,  or  in  fractional  numbers. 
For  example,  43  is  not  a  cubic  number ;  I  say  therefore  that  it 
is  impossible  to  assign  any  number,  either  integer  or  fractional, 
whose  cube  shall  be  exactly  43.  We  may  however  affirm,  that 
tlie  cube  root  of  that  number  is  greater  than  3,  since  the  cube 
of  3  is  only  27 ;  and  less  than  4,  because  the  cube  of  4  is  64. 
We  know  therefore,  that  the  cube  root  required  is  necessarily 
contained  between  the  numbers  3  and  4. 

161.  Since  the  cube  root  of  43  is  greater  than  3,  if  we  add  a 
fraction  to  3,  it  is  certain  that  we  may  approximate  still  nearer 
and  nearer  to  the  true  value  of  this  root ;  but  we  can  never 
assign  the  number  which  expresses  that  value  exactly  ;  because 
the  cube  of  a  mixt  number  can  never  be  perfectly  equal  to  an 
integer,  such  as  43.  If  we  were  to  suppose,  for  example,  3|,  or 
I  to  be  the  cube  root  required,  the  error  would  be  -J  ;  for  the 
cube  of  {  is  only  ^i^,  or  42|. 

162.  This  therefore  shews,  that  the  cube  root  of  45  cannot  he 
expressed  in  any  way,  either  hy  integers  or  by  fractions.  How- 
ever we  have  a  distinct  idea  of  the  magnitude  of  this  root ; 

3 

which  induces  us  to  use,  in  order  to  represent  it,  the  sign  ^, 
which  we  place  before  the  proposed  number,  and  which  is  read 
cube  rootf  to  distinguish  it  from  the  square  root,  which  is  often  called 

3 

simply  the  root.  Thus  v'43  means  the  cube  root  of  43,  that  is  to 
say,  tlie  number  whose  cube  is  43,  or  which,  multiplied  thrice 
by  itself,  produces  43. 


44  dlgebra.  Sect  1. 

163.  It  is  evident  also,  that  such  expressions  cannot  belong 
to  rational  quantities,  and  that  they  rather  form  a  particular 
species  of  irrational  quantities.  They  have  nothing  in  common 
with  square  roots,  and  it  is  not  possible  to  express  such  a  cube 
root  by  a  square  root ;  as,  for  example,  by  v^i2  ;  for  tlie  square 
of  vr2  being  12,  its  cube  will  be  12  v/i2>  consequently  still  irra- 
tional, and  such  cannot  be  equal  to  43. 

164.  If  the  proposed  number  be  a  real  cube,  our  expressions 
become  rational ;  v^r  is  equal  to  1  ;  ys"  is  equal  to  2;  ^^  is 
equal  to  3  ;  and,  generally,  Vaaa  is  equal  to  a. 

156.  If  it  were  proposed  to  multiphj  one  mbe  root,  ^/^  by  another, 

Vb,  the  product  must  be  ^/ab  ;  for  we  know  that  the  cube  root  of 
a  product  ab  is  found  by  multiplying  together  the  cube  roots  of 

3  3 

the  factors.     Hence,  also,  ij  we  divide  ^a  by  ^V,  the  qu-otient 

will  be    fi.. 
\b 

3 a 

166.  We  furtlier  perceive,  that  2  \/a,  is  equal  to  \/iZ,  because 

3  _  3  _  3  _  3 

2  is  equivalent  to  x/8  -,   that  3  \/a   is  equal  \/27a,  and  b  ^a  is 

3 

equal  to  \/abbb.  So,  reciprocally,  if  the  number  under  the  radi- 
cal sign  has  a  factor  which  is  a  cube,  we  may  make  it  disappear 
by  placing  its  cube  root  before  the  sign.    For  example,  instead 

3_  3_  '3_  3 

of  \/64a  we  may  write  4  \/a ;   and   5  \/a  instead  of  \/i25a. 

3  _  3  _ 

Hence  v^ie  is  equal  to  2  \/2,  because  16  is  equal  to  8  x  2. 

167.  When  a  number  proposed  is  negative,  its  cube  root  is 
not  subject  to  the  same  difficulties  that  occurred  in  treating  of 
square  roots.  For,  since  the  cubes  of  negative  numbers  are 
negative,  it  follows  that  the  cube  roots  of  negative  numbers  are 

3    3  _ 

only  negative.     Thus,  v— ^  i*5  equal  to  —  2,  and  V— 27  to  — •  3. 

3 3 3_ 

It  follows  also,  that  v^— 12  is  the  same  as  —  vi2>  and  that  \/—a 

3  

may  be  expressed  by  —  \/a»  Whence  we  see,  that  the  sign  — , 
when  it  is  found  after  the  sign  of  the  cube  root,  might  also  have 
been  placed  before  it.  We  are  not  therefore  led  here  to  impos- 
sible, or  imaginary  numbers,  which  happened  in  considering  the 
square  roots  of  negative  numbers. 


Chap,  16.  Of  Simple  Quantities,  45 

CHAPTER  XVI. 

(ff  Powers  in  general, 

168.  The  product,  which  we  obtain  by  multiplying  a  mimber 
several  times  by  itself,  is  called  a  power.  Thus,  a  square  which 
arises  from  the  multiplication  of  a  iiumher  by  itself,  and  a  cube 
which  we  obtain  by  multiplying  a  number  thrice  by  itself,  are 
powers.  TVe  say  also  in  the  former  case,  that  the  number  is  raised 
to  the  second  degree,  or  to  tJie  second  power  ;  and  in  the  latter,  that 
the  number  is  raised  to  the  third  degree,  or  to  the  third  power. 

169.  We  distinguish  those  powers  from  one  another  by  the 
mimber  of  times  that  the  given  number  has  been  multiplied  by 
itself.  For  example,  a  square  is  called  the  second  power, 
because  a  certain  given  number  has  been  multiplied  twice  by 
itself ;  and  if  a  number  has  been  multiplied  thrice  by  itself,  we 
call  the  product  the  third  power,  which  therefore  means  the 
same  as  the  cube.  Multiply  a  number  four  times  by  itself,  and 
you  will  have  its  fourth  power,  or  what  is  commonly  called  the 
hi-quadrate.  From  what  has  been  said  it  will  be  easy  to  under- 
stand what  is  meant  by  the  fifth,  sixth,  seventh,  &c.  power  of  a 
number.  I  only  add,  that  the  names  of  these  powers,  after  the 
fourth  degree,  cease  to  have  any  other  but  these  numeral  dis- 
tinctions. 

170.  To  illustrate  this  still  better,  we  may  observe,  in  the 
first  place,  that  the  powers  of  1  remain  always  the  same  ;  because, 
whatever  number  of  times  we  multiply  1  by  itself,  the  product 
is  found  to  be  always  1 .  We  shall  therefore  begin  by  repre- 
senting the  powers  of  2  and  of  3.  They  succeed  in  the  following 
order  j 


46 


Algebra, 


Sect.  1. 


Powers. 


Ofthe  number  2. 

' — -^ — 


Of  the  number  3. 

V i 


2  3 

4  9 

8  27 

16  81 

32  243 

64  729 

128  2187 

256  6561 

512  19683 

1024  59049 

2048  177147 

4096  531441 

8192  1594323 

16384  4782969 

32768  14348907 

65536  43046721 

131072  129140163 

262144  387420489 

But  the  powers  of  the  number  10  are  the  most  remarkable ; 
for  on  these  powers  the  system  of  our  arithmetic  is  founded,  A 
few  of  them  ranged  in  order,  and  beginning  with  the  first  power, 
are  as  follows : 

I.        11.        III.        IV.  V.  VI. 

10,        100,       1000,     10000,        100000,         1000000,  6lC. 

171.  In  order  to  illustrate  this  subject,  and  to  consider  it  in  a 
more  general  manner,  we  may  observe,  that  the  powers  of  any 
number,  a,  succeed  each  other  in  the  following  order  : 

I.     II.    III.      IV.         V.  VI. 

a,  aa,  aaa,  aaaa,  aaaaa,  aaaaaa,  &c. 
But  we  soon  feel  the  inconvenience  attending  this  manner  of 
writing  the  powers,  which  consists  in  the  necessity  of  repeating 
the  same  letter  very  often,  to  express  high  powers ;  and  the 
reader  also  would  have  no  less  trouble,  if  he  were  obliged  to 
count  all  the  letters,  to  know  what  power  is  intended  to  be 
represented.  The  hundredth  power,  for  example,  could  not  be 
conveniently  written  in  this  mariner ;  and  it  would  be  still  more 
difficult  to  read  it. 

172.  To  avoid  this  inconvenience,  a  much  more  commodious 
method  of  expressing  such  powers  has  been  devised,  which  from 


I 


Cliap.  16.  Of  Simple  Quantities.  47 

its  extensive  use  deserves  to  be  caFefully  explained;  vi%.  To 
express,  for  example,  the  hundredth  power,  we  simply  write  the 
number  100  above  the  number  whose  hundredth  power  we  would 
express,  and  a  little  towards  the  right-hand;  thus  a* ««  means 
a  raised  to  100,  and  represents  the  hundreth  power  of  a.  It  must 
be  observed,  that  the  name  exponent  is  given  to  the  number  writ- 
ten  above  that  whose  power,  or  degree,  it  represents,  and  which  in 
the  present  instance  is  100. 

173.  In  the  same  manner,  o^  signifies  a  raised  to  2,  or  the 
second  power  of  a,  which  we  i^present  sometimes  also  by  aa, 
because  both  these  expressions  are  written  and  understood  with 
equal  facility.  But  to  express  the  cube,  or  the  third  power  aaa, 
we  write  a^  according  to  the  rule,  that  we  may  occupy  less  room. 
So  a*  signifies  the  fourth,  a*  the  fifth,  and  a«  the  sixth  power 
of  a. 

174.  In  a  word,  all  the  powers  of  a  will  be  represented  by  a, 
a^9  a^fa^,  a*,  a®,  a'' ,  a®,  a',  a*°,  &c.  Whence  we  see  that  in 
this  manner,  we  might  very  properly  have  written  a^  instead 
of  a  for  the  first  term,  to  shew  the  order  of  the  series  more 
clearly.  In  fact  a*  is  no  more  than  a,  as  this  unit  shews  that  the 
letter  a  is  to  be  written  only  once.  Such  a  series  of  powers  is 
called  also  a  geometrical  progression,  because  each  term  is 
greater  by  one  than  the  preceding. 

175.  As  in  this  series  of  powers  each  term  is  found  by  multi- 
plying the  preceding  term  by  a,  which  increases  the  exponent 
by  1  ;  so  when  any  term  is  given,  we  may  also  find  the  preced- 
ing one,  if  we  divide  by  a,  because  this  diminishes  the  exponent 
by  1.  This  shews  that  the  term  which  precedes  the  first  term  a* 

must  necessarily  be  — ,  or  1 ;  now,  if  we  proceed  according  to 

the  exponents,  we  immediately  conclude,  that  the  term  which 
precedes  the  first  must  be  a^.  Hence  we  deduce  this  remark- 
able property ;  that  a°  is  constantly  equal  to  1,  however  great  or 
small  the  value  of  the  number  a  may  be,  and  even  w  hen  a  is  noth- 
ing ;  that  is  to  say,  a^  is  equal  to  1. 

176.  We  may  continue  our  series  of  powers  in  a  retrograde 
order,  and  that  in  two  different  ways  ;  first,  by  dividing  always 
by  a,  and  secondly  by  diminishing  the  exponent  by  unity.    And 


48 


Mgehra, 


Sect,  1* 


it  is  evident  that,  whether  we  follow  the  one  or  the  other,  the 
terms  are  still  perfectly  equal.  This  decreasing  series  is 
represented,  in  both  forms,  in  the  following  table,  which  must 
be  read  backwards,  or  from  right  to  left. 


1 

1 

1 

1 
aaa 

1 

1 

1 

a 

aaaaaa 

aaaaa 

aaaa 

aa 

a 

1 

a} 

1 
a' 

1 

1 
a' 

1 

1 

a-6 

a   « 

a-4 

a-3 

a-2 

a-i 

a« 

a^ 

1. 

2. 


177.  We  have  come  now  to  the  knowledge  of  powers,  whose 
exponents  are  negative,  and  are  enabled  to  assign  the  precise 
value  of  those  powers.  From  what  has  been  said,  it  appears 
that> 

f  1 ;  then 

1 
a' 

1 

aa 


>  is  equal  to  < 


1 

or—. 


1-4 


ra  ' 


h 


&c. 


178.  It  will  be  easy,  from  the  feregoing  notation,  to  find  the 
powers  of  a  product,  ah.  They  must  evidently  be  ah,  or  a*b', 
a*b*,  a^b*,  a'*b'*,  a*b*,  ^c.  And  the  powers  of  fractions  will  be 
found  in  tJie  same  manner  ;  for  example  those  of  -^  are. 


a^    2l^    a^    a'*    a^    a^    a^ 

bl'  b^'  b^'  b^'  b^'  be'  b7' 


&c. 


179.  Lastly,  we  have  to  consider  the  powers  of  negative  num- 
bers. Suppose  the  given  number  to  be  —  a  5  its  powers  will 
form  the  following  series  : 

— -  a,  +  aa,  —  a^,  +  a^,  •—  a^,  +  a^,  &c. 


Chap.  17.  Of  Simple  quantities.  49 

We  may  observe,  that  those  powers  only  become  negative, 
whose  exponents  are  odd  numbers,  and  that,  on  the  contrary, 
all  the  powers,  which  have  an  even  number  for  the  exponent, 
are  positive.  So  that,  the  third,  fifth,  seventh,  ninth,  &c. 
powers  have  all  the  sign  — ;  and  the  second,  fourth,  sixth, 
eighth,  &c.  powers  are  affected  with  the  sign  +. 


CHAPTER  XVII. 

Of  the  calculati{)n  of  Powers. 

180.  We  have  nothing  in  particular  to  observe  with  regard 
to  the  addition  and  subtraction  of  powers  ;  for  we  only  repre- 
sent these  operations  by  means  of  the  signs  -|-  and  — ,  when  the 
powers  are  different.  For  example,  a'  -f  a*  is  the  sum  of  the 
second  and  third  powers  of  a ;  and  a*  — ■  a^  is  what  remains 
when  we  subtract  the  fourth  power  of  Sifrom  the  fifth  ;  and  neither 
of  these  results  can  he  abridged.  When  we  have  powers  of  the 
same  kind,  or  degree,  it  is  evidently  unnecessary  to  connect 
them  by  signs  ;  a^  +  a*  makes  2a 3,  &c. 

181.  But,  in  the  multiplication  of  powers,  several  things 
require  attention. 

First,  when  it  is  required  to  multiply  any  power  of  a  by  a, 
we  obtain  the  succeeding  power ;  that  is  to  say,  the  power  whose 
exponent  is  greater  by  one  unit.  Thus  a*,  multiplied  by  a, 
produces  a^  ;  and  a'  multiplied  by  a,  produces  a*.  And,  in 
the  same  manner,  when  it  is  required  to  multiply  by  a  the 
powers  of  that  number  which  have  negative  exponents,  we  must 
add  1  to  the  exponent.  Thus,  a~i  multiplied  by  a  produces  a'*  or 
1  ,•  which  is  made  more  evident  by  considering  that  a~*  is  equal 

to  — ,  and  that  the  product  of  —  by  a  being  — ,  it  is  consequently 

equal  to  1.     Likewise  a"^  multiplied  by  a,  produces  a"^,  or 

—  ;  and  a~^°,  multiplied  by  a,  gives  a~^,  and  so  on. 
(*■ 

182.  Next,  if  it  be  required  to  multiply  a  power  of  a  by  aa, 
or  the  second  power,  I  say  that  the  exponent  becomes  greater 
by  2.    Thus,  the  product  of  a*  by  a*  is  a*  ;  that  of  a*  by  a^  is 

7 


50  Mgebra.  Sect.  1. 

ft*  ;  that  of  a*  by  a*  is  a®  ;  and,  more  generally  a"  multiplied 
by  a^  makes  a"+2.  With  regard  to  negative  exponents,  we  shall 
have  a*,  or  a^for  the  product  of  sr^  by  'd^  ;  for  a"^  being  equal 

to  — ,  it  is  the  same  as  if  we  had  divided  aa  by  a  ;  consequently 

the  product  required  is  — ,   or  a.     So  a"^,  multiplied  by  a', 

produces  a<>,  or  1  ;  and  a"^,  midtiplied  by  a*,  produces  a""^. 

183.  It  is  no  less  evident  that,  to  multiply  any  power  of  a  by 
G^,  we  must  increase  its  exponent  by  three  units ;  and  tliat 
consequently  the  product  of  a"  by  a^  is  a"+^.  And  whenevr  it 
is  required  to  midtiply  together  two  powers  of  a,  the  product  xvill 
he  also  a  power  of  a,  and  a  power  wlwse  exponent  will  be  the  sum 
of  those  of  the  two  given  powers.  For  example,  ft*  multiplied  by 
a*  will  make  a^,  and  a^  ^  multiplied  by  a''  will  produce  ft^  ®,  kc» 

184.  From  these  considerations  we  may  easily  determine  the 
highest  powers.  To  find,  for  instance,  the  twenty -fourth  power 
of  2,  I  multiply  the  twelfth  power  by  the  twelfth  power,  because 
2*"*  is  equal  to  2^2  >^  2^*.  Now  we  have  already  seen  that 
2**  is  4096  ;  I  say  therefore  that  the  number  16777216,  or  the 
product  of  4096  by  4096,  expresses  the  power  required,  2^  *, 

185.  Let  us  proceed  to  division.  We  shall  remark  in  the 
first  place,  that  to  divide  a  power  ojaby  2l,  we  must  subtract  1 

from  tlie  exponent,  or  diminish  it  by  unity.    Thus  a*,  divided  by 

ft,  gives  a^  ;  ft 0,  or  1,  divided  by  a,  is  equal  to  ft""*  or  —  ;  ft-^, 

divided  by  a,  gives  ft"-*. 

186.  If  we  have  to  divide  a  given  power  of  a  by  ft*,  we  must 
diminish  the  exponent  by  2 ;  and  if  by  a^,  we  must  subtract 
three  units  from  the  exponent  of  the  power  proposed.  So,  in 
general,  whatercer  power  of  2i  it  is  required  to  divide  by  another 
power  0^  a,  the  rule  is  always  to  subtract  the  exponent  of  the  se<;o7id 
from  the  exponent  of  the  first  of  these  powers.  Thus  a^^,  divided 
by  ft%  will  give  a^  ;  ft®,  divided  by  ft^,  will  give  a~^  ;  and  a'^, 
divided  by  ft'*,  will  give  a~^. 

187.  From  what  has  been  said  above,  it  is  easy  to  understand 
the  method  of  finding  the  powers  of  powers,  this  being  done  by 
multiplication.  When  we  seek,  for  example,  the  square,  or  the 
second  power  of  a^,  we  find  (i-^  ;  and  in  the  same  manner  we 


Chap.  18.  Of  Simple  Quantities.  51 

find  a»*  for  the  third  power,  or  the  cuhe  of  a*.  To  obtain  the 
square  of  a  power ,  we  have  oidy  to  d&iible  its  exponent ;  for  its  cuhe, 
we  must  triple  the  expment;  and  so  on.  The  square  of  a"  is 
a*";  the  cube  of  a"  is  a^";  the  seventh  power  of  a"  is  a^",  &c. 

188.  The  square  of  a',  or  the  square  of  the  square  of  a, 
being  a^,  we  see  why  the  fourtli  power  is  called  the  bi-quadrate. 
The  square  of  a^  is  a®  ;  the  sixth  power  has  therefore  received 
the  name  of  the  square-cubed. 

Lastly,  the  cube  of  a^  being  a^,  we  call  the  ninth  power  the 
cuho-cuhe.  No  other  denominations  of  this  kind  have  been 
introduced  for  powers,  and  indeed  the  two  last  are  very  little 
used. 


CHAPTER  XVIII. 

Of  Roots  with  relation  to  Powers  in  general. 

189.  Since  the  square  root  of  a  given  number  is  a  number, 
whose  square  is  equal  to  that  given  number ;  and  since  the  cube 
root  of  a  given  number  is  a  number,  whose  cube  is  equal  to  that 
given  number ;  it  follows  that  any  number  whatever  being  given, 
we  may  always  indicate  such  roots  of  it,  that  their  fourth,  or 
their  fifth,  or  any  other  power,  may  be  equal  to  the  given  num- 
ber. To  distinguish  these  diiferent  kinds  of  roots  better,  we 
shall  call  the  square  root,  the  second  root ;  and  the  cube  root,  the 
third  root;  because  according  to  this  denomination,  we  may  call 
the  fourth  root^  that  whose  biquadrate  is  equal  to  a  given  num- 
ber ;  and  the  fifth  root,  that  whose  fifth  power  is  equal  to  a  given 
number,  &c. 

190.  As  the  square,  or  second  root,  is  marked  by  the  sign 

v/,  and  the  cubic  or  third  root  by  the  sign  \^,  so  the  fourth  root 

is  represented  by  the  sign  \/  ;  the  fifth  root  by  the  sign  -v/ ;  and 
so  on ;  it  is  evident  that  according  to  this  method  of  expression, 

2 

the  sign  of  the  square  root  ought  to  be  \/.  But  as  of  all  roots 
this  occurs  most  frequently,  it  has  been  agreed,  for  the  sake  of 
brevity,  to  omit  the  number  2  in  the  sign  of  this  root.     So  that 


52 


Mgebra. 


&ect.  1. 


when  a  radical  sign  has  no  number  prefixed,  this  always  shews 
that  the  square  root  is  to  be  understood, 

191.  To  explain  this  matter  still  further,  we  shall  here  exhibit 
the  different  roots  of  the  number  a,  with  their  respective  values  : 


Va  "I  fSd  ^ 

3  

V«  3d 

V«  r*  is  the  <^  4th 


a, 

a, 
>root  of  <^  a. 


s 


s/a    J 
So  that  conversely ; 
The  2d 


5th 
6th 


The  3d 
The  4th 
The  5th 


3  _ 


>  power  of  < 


a,  and  so  on. 


a, 
a. 


s/a    )>  is  equal  to  -{  a, 
a, 


s  _ 
\/a 


Va   J 


La, 


The  6th 
and  so  on. 

192.  Whether  the  number  a  therefore  be  great  or  small,  we 
know  what  value  to  affix  to  all  these  roots  of  different  degrees. 

It  must  be  remarked  also,  that  if  we  substitute  unity  for  a,  all 
those  roots  remain  constantly  1  ;  because  all  the  powers  of  1 
have  unity  for  their  value.  If  the  number  a  be  greater  than  1, 
all  its  roots  will  also  exceed  unity.  Lastly,  if  that  number  be 
less  than  1,  all  its  roots  will  also  be  less  than  unity. 

193.  When  the  number  a  is  positive,  we  know  from  what  was 
before  said  of  the  square  and  cube  roots,  that  all  the  other  roots 
may  also  be  determined,  and  will  be  real  and  possible  numbers. 

But  if  the  number  a  is  negative,  its  second,  fourth,  sixth,  and 
all  even  roots,  become  impossible,  or  imaginary  numbers ; 
because  all  the  even  powers,  whether  of  positive,  or  of  negative 
numbers,  are  affected  with  the  sign  +.  Whereas  the  third,  fifth, 
seventh,  and  all  odd  roots,  become  negative,  but  rational ;  because 
the  odd  powers  of  negative  numbers,  are  also  negative. 


.Chap.  19.  Of  Simple  Quantities,  53 

194.  We  have  here  also  an  inexhaustible  source  of  new  kinds 
of  surd,  or  irrational  quantities ;  for  whenever  the  number  a  is 
not  really  such  a  power,  as  some  one  of  the  foregoing  indices 
represents,  or  seems  to  require,  it  is  impossible  to  express 
that  root  either  in  whole  numbers  or  in  fractions ;  and  conse- 
quently it  must  be  classed  among  the  numbers  which  are  called 
irrational. 


CHAPTER  XIX. 

Of  the  Method  of  representing  Irrational  J^umhers  hy  Fractional 

Exponents, 

195.  We  have  shewn  in  the  preceding  chapter,  that  the  square 
of  any  power  is  found  by  doubling  the  exponent  of  that  power, 
and  that  in  general  the  square,  or  the  second  power  of  a",  is 
a'".  The  converse  follows,  viz.  that  the  square  root  of  the  power 
a*°  is  fl",  and  that  it  is  found  hy  taking  half  the  exponent  of  that 
power,  or  dividing  it  by  2. 

196.  Thus  the  square  root  of  a^  is  aM  that  of  a"^  is  a*  . 
that  of  a"  is  a^  ;  and  so  on.     And  as  this  is  general,  the  square 

S  5 

root  of  a^  must  necessarily  be  a^  and  that  of  a*  d^.     Con- 

sequently  we  shall  have  in  the  same  manner  d^  for  the  square 

root  of  a^  ;  whence  we"  see  that  a^  is  equal  to  ^a.  ;  and  this 
new  method  of  representing  the  square  root  demands  particular 
attention. 

197.  We  have  also  shown  that,  to  find  the  cube  of  a  power  as 
«",  we  must  multiply  its  exponent  by  3,  and  that  consequently 
that  cube  is  a^". 

So  conversely,  when  it  is  required  to  find  the  third  or  cube 
root,  of  the  power  a^",  we  have  only  to  divide  that  exponent  by 
3,  and  may  with  certainty  conclude,  that  the  root  required  is  a". 
Consequently  a^,  or  a,  is  the  cube  root  of  a*  ;  a*  is  that  of  a^  ; 
a 3  is  that  of  a^  ;  and  so  on. 

198.  There  is  nothing  to  prevent  us  from  applying  the  same 
reasoning  to  those  cases  in  which  the  exponent  is  not  divisible 


54  Mgebm,  Sect.  1. 

by  3,  and  concluding  that  the  cube  root  of  a*  is  a^,  and  that  the 
cube  root  of  a*  is  a^,  or  a^^.  Consequently  the  third,  or 
cube  root  of  a  also,  or  o*,  must  be  a'.  Whence  it  appears  that 
a^  is  equal  to  ^Z 

199.  It  is  the  same  with  roots  of  a  higher  degree.     The 
fourth  root  of  a  will  be  a*,  which  expression  has  the  same  value 

4  1 

as  v«.     The  fifth  root  of  a  will  be  a^,  which  is  consequently 

equivalent  to  -y/^;  and  the  same  observation  may  be  extended 
to  all  roots  of  a  higher  degree. 

200.  We  might  therefore  entirely  reject  the  radical  signs  at 
present  made  use  of,  and  employ  in  their  stead  the  fractional 
exponents  which  we  have  explained  ;  however,  as  we  have  been 
long  accumstomed  to  those  signs,  and  meet  with  them  in  all 
books  of  algebra,  it  would  be  wrong  to  banish  them  entirely 
from  calculation.  But  there  is  sufficient  reason  also  to  employ, 
as  is  now  frequently  done,  the  other  method  of  notation,  because 
it  manifestly  corresponds  with  the  nature  of  the  thing.   In  fact, 

we  see  immediately  that  o^  is  the  square  root  of  a,  because  we 

know  that  the  square  of  a',  that  is  to  say,   a^  multiplied  by 

fl"*,  is  equal  to  a*  or  a, 

201.  What  has  now  been  said  is  sufficient  to  shew  how  we 
are  to  understand  all  other  fractional  exponents  that  may  occur. 

4 

If  we  have,  for  example,  a^,  this  means  that  we  must  first 
take  the  fourth  power  of  a,  and  then  extract  its  cube  or  third 

4  S  _ 

root;    so  that  a"*   is  the  same  as  the  common  expression,  Va-*. 

3 

To  find  the  value  of  a^,  we  must  first  take  the  cube,  or  the 
third  power  of  o,  which  is  a^,  and  then  extract  the  fourth  root 

3^  4  4 

of  that  power;   so  that  a*  is  the  same  as    v^as.      So,   aJ  is 

equal  to  \/^4.,  &c. 

202.  When  the  fraction  which  represents  the  exponent  ex- 
ceeds unity,  we  may  express  the  value  of  the  given  quantity  in 

5 

another  way.  Suppose  it  to  be  a^  ;  this  quantity  is  equivalent 
to   a  ^,  which  is  the  product  of  a*   by  aJ,     Now  a^  being 


Chap.  19.  Of  Simple  ^lantities,  55 

equal  to  \/a,  it  is  evident  that  a^  is  equal  to  a*  \/a.     So  o  ^  , 

31  3_  15  33 

or  a  ^  is  equal  to  a^  \/a    and  a  ^ ,    that  is    a  ^,    expresses 

a^  V^.     These  examples  are  sufficient  to  illustrate  the  great 
utility  of  fractional  exponents. 
203.  Their  use  extends  also  to  fractional  numbers  :  let  there 

be  given  — =,  we  know  that  this  quantity  is  equal  to  -^  5    now 

we  have  seen  already  that  a  fraction  of  the  form  —  may  be  ex- 
pressed by  a""" ;  so  instead  of  — t=  we  may  use  the  expression 

\/a 
1  11 

a~^.     In  the  same  manner,  - —  is  equal  to  or^.      Again,    let 


be  propose 


fl2 

the  quantity  - —  be  proposed ;  let  it  be  transformed  into  this. 


3  / 

"^f  which  is  the  product  of  a'  by  a-*  ;  now  this  product  is  equi- 
a* 

valent  to  a"^,  or  to  a  ^,  or  lastly  to  \/a  .  Practice  will  ren- 
der similar  reductions  easy. 

204.  We  shall  observe,  in  the  last  place,  that  each  root  may 

be  represented  in  a  variety  of  ways.    For  >/^  being  the  same 

1 
as   a^,  and  ^  being  transformable  into  all  these  fractions,  |,  |,i, 

tV»  TS9  ^^'  i*  is  evident  that  v/a  is  equal  to  ^as,  and  to  ^/a^  and 

8  _  3   _ 

to  v/a4  and  so  on.     In  the  same  manner  y/a   which  is  equal 

to  a^,  will  be  equal  to  -v/as,  and  to  v^^j  and  to  \/a4>.  And 
we  see  also,  that  the  number  a,  or  aS  might  be  represented  by 
the  following  radical  expressions  : 

S_        3_         4_        5__ 

\/a2j   V*^'    \/'a4>9  \/aSf  &C. 

205.  This  property  is  of  great  use   in  multiplication   and 

,  2  _  3_ 

division  :  for  if  wc  have,  for  example,  to  multiply  \/a   by  \/a, 

6  _  3 6  3    

we  write  ^as,  for  \/a,  and  \/a3  instead  of  >/«/  i^  this 
manner  we  obtain  the  same  radical  sign  for  both,  and  the  mul- 

6 

tiplication  being  now  performed,  gives  the  product  \/as»     The 

111  1 

same  result  is  deduced  from  a^'^^,  the  product  of  n^  multi- 


56  Mgebra.  Sect.  I, 


plied  by   a^ ;  for  J  +  |  is  f ,  and  consequently  the  product 
required  is  a*  or  Va«. 

9  1  3  1 

If   it    were    required   to  divide  -y/a,  or  a»,  by  v^a,  or  a"^,  we 

I  1  3  _^  2 

should  have  for  the  quotient  a*      "^^   or  a^      ^,   that  is  to  say 
a^,  or  ^Z 


CHAPTER  XX. 

Of  the  different  Methods  of  Calculation,  and  of  their  mutual 
Connexion, 

206.  Hitherto  we  have  only  explained  the  different  methods 
of  calculation  :  addition,  subtraction,  multiplication,  and  divis- 
ion ;  the  involution  of  powers,  and  the  extraction  of  roots.  It 
will  not  be  improper  therefore,  in  this  place,  to  trace  back  the 
origin  of  these  different  methods,  and  to  explain  the  connexion 
which  subsists  among  them  ;  in  order  that  we  may  satisfy  our- 
selves whether  it  be  possible  or  not  for  other  operations  of  the 
same  kind  to  exist.  This  inquiry  will  throw  new  light  on  the 
subjects  which  we  have  considered. 

In  prosecuting  this  design,  we  shall  make  use  of  a  new  cha- 
racter, which  may  be  employed  instead  of  the  expression  that 
has  been  so  often  repeated,  is  equal  to  ;  this  sign  is  =,  and  is 
read  is  equal  to.  Thus,  when  I  write  a=  b,  this  means  that  a 
is  equal  to  b :  so,  for  example  3x5  =  15. 

207.  The  first  mode  of  calculation  which  presents  itself  to  the 
mind,  is  undoubtedly  addition,  by  which  we  add  two  numbers 
together  and  find  their  sum.  Let  a  and  b  then  be  the  two  given 
numbers,  and  let  their  sum  be  expressed  by  the  letter  c,  we  shall 
have  a  -f-  &  =  c.  So  that  whe^i  we  know  the  two  numbers  a  and 
5,  addition  teaches  us  to  find  the  number  c, 

208.  Preserving  this  comparison  a  +  b=  c,  let  us  reverse  the 
question  by  asking,  how  we  are  to  find  the  number  6,  when  we 
know  the  numbers  a  and  c. 

It  is  required  therefore  to  know  what  number  must  be  added 
to  a,  in  order  that  the  sum  may  be  the  number  c.  Suppose,  for 
example,  a  =  3  and  c  =  8 ;  so  that  we  must  have  3  -|-  6  =  8 ;  b 


Chap.  20.  Of  Simple  ^lantities,  5T 

will  evidently  be  found  by  subtracting  3  from  8.   So,  in  general, 
to  find  bf  we  must  subtract  a  from  c,  whence  arises  6  =  c  —  a; 
for  by  adding  a  to  both  sides  again,  we  have  6-|-a=c  —  a-fa, 
that  is  to  say  =  c,  as  we  supposed. 
Such  then  is  the  origin  of  subtraction. 

209.  Subtraction  therefore  takes  place,  when  we  invert  the 
question  which  gives  rise  to  addition.  Now  the  number  which 
it  is  required  to  subtract  may  happen  to  be  greater  than  that 
from  which  it  is  to  be  subtracted ;  as,  for  example,  if  it  were 
required  to  subtract  9  from  5  :  this  instance  therefore  furnishes 
us  with  the  idea  of  a  new  kind  of  numbers,  which  we  call  nega- 
tive numbers,  because  5  —  9  =  — '4. 

210.  When  several  numbers  are  to  be  added  together  which 
are  all  equal,  their  sum  is  found  by  multiplication,  and  is  called 
a  product.  Thus  ab  means  the  product  arising  from  the  multi- 
plication of  a  by  b,  or  from  the  addition  of  a  number  a  to  itself 
b  number  of  times.  If  we  represent  this  product  by  the  letter 
c,  we  shall  have  ab  =.c;  and  multiplication  teaches  us  how  to 
determine  the  number  c,  when  the  numbers  a  and  b  are  known. 

211.  Let  us  now  propose  the  following  question  :  the  numbers 
a  and  c  being  known,  to  find  the  number  b,  Supjjose  for 
example,  a  =  3  and  c  =  15,  so  that  S&  =15,  w^e  ask  by  what 
number  3  must  be  multiplied,  in  order  that  the  product  may  be 
15  :  for  the  question  proposed  is  reduced  to  this.  Now  this  is 
division :  the  number  required  is  found  by  dividing  15  by  3  ; 
and  therefore,  in  general,  the  number  b  is  found  by  dividing  c 

by  a  ;  from  which  results  the  equation  5  =  — . 

212.  Now,  as  it  frequently  happens  that  the  number  c  cannot 
be  really  divided  by  the  number  «,  while  the  letter  b  must  how- 
ever have  a  determinate  value,  another  new  kind  of  numbers 
presents  itself;  these  are  fractions.  For  example,  supposing 
a  =  4,  c=  3,  so  that  46  =  3,  it  is  evident  that  b  cannot  be  an 
integer,  but  a  fraction,  and  that  we  shall  have  6  =  |. 

213.  We  have  seen  that  multiplication  arises  from  addi- 
tion, tliat  is  to  say,  from  the  addition  of  several  equal 
quantities.  If  we  now  proceed  further,  we  shall  perceive 
that  from  the  multiplication   of  several   equal  quantities  to- 


5a  Mgebra.  Sect.  1. 

gether  powei*s  ai'e  derived.  Those  powers  are  represented  in 
a  general  manner  by  tlie  expression  a\  which  signifies  that  the 
number  a  must  be  multiplied  as  many  times  by  itself  as  is 
denoted  by  the  number  b.  And  we  know  from  what  has  been 
already  said,  that  in  the  present  instance  a  is  called  the  root,  b 
the  exponent,  and  a*  the  power. 

214.  Further,  if  we  represent  this  power  also  by  the  letter  c, 
we  have  a*  =  c,  an  equation  in  which  three  letters  a,  b,  c,  are 
found.  Now  we  have  shewn  in  treating  of  powers,  how  to  find 
the  power  itself,  that  is,  the  letter  c,  when  a  root  a  and  its 
exponent  b  are  given.  Suppose,  for  example,  a  =5,  and  6=3, 
so  that  cz=5^  ;  it  is  evident  that  we  must  take  the  third  power 
of  5,  which  is  125,  and  that  thus  c=  125. 

215.  \Ye  have  seen  how  to  determine  the  power  c,  by  means 
of  the  root  a  and  the  exponent  6;  but  if  we  wish  to  reverse  the 
question,  we  shall  find  that  this  may  be  done  in  two  ways,  and 
that  there  are  two  different  cases  to  be  considered  :  for  if  two 
of  these  three  numbers  a,  &,  c,  were  given,  and  it  were  required 
to  find  the  third,  we  should  immediately  perceive  that  this 
question  admits  of  tliree  different  suppositions^  and  consequently 
three  solutions.  We  have  considered  the  case  in  which  a  and  6 
were  the  numbers  given,  we  may  therefore  suppose  further  that 
c  and  fl,  or  c  and  b  are  known,  and  that  it  is  required  to  deter- 
mine the  third  letter.  Let  us  point  out  therefore,  before  we 
proceed  any  further,  a  very  essential  distinction  between  invo- 
lution and  the  two  operations  which  lead  to  it.  When  in 
addition  we  reversed  the  question,  it  could  be  done  only  in  one 
way ;  it  was  a  matter  of  indifference  whether  we  took  c  and  a, 
or  c  and  b,  for  the  given  numbers,  because  we  might  indiffer- 
ently write  a  -f-  6,  or  &  -f  a.  It  was  the  same  with  multiplica- 
tion ;  we  could  at  pleasure  take  the  letters  a  and  b  for  each 
other,  the  equation  ab  =  c  being  exactly  the  same  as  ba  =  c. 

In  the  calculation  of  powers,  on  the  contrary,  the  same  thing 
does  not  take  place,  and  we  can  by  no  means  write  b°  instead  of 
a*.  A  single  example  will  be  sufficient  to  illustrate  this  :  let  a 
=  5,  and  6  =  3;  we  have  a!'  z=z5^  =z  125.  But  6«  =  3*  =  243  : 
two  very  different  results. 


SECTION  SECOND. 

•F  THE  DIFFERENT  METHODS  OF  CALCULATING  COMPOUND  QUANTITIEgv 

CHAPTER  I. 

Of  the  Mdition  of  Compound  ^antities.  itAii 

ARTICLE    216. 

When  two  or  more  expressions,  consisting  of  several  terms, 
are  to  be  added  together,  the  operation  is  frequently  represented 
merely  by  signs,  placing  each  expression  between  two  paren- 
theses, and  connecting  it  with  the  rest  by  means  of  tl\e  sign  +. 
If  it  be  required,  for  example,  to  add  the  expressions  o  +  6  -f  c 
and  d  -f-  e  +/,  we  represent  the  sum  thus  : 
(a  +  6  +  c)  +  (fi  +  e  +/ 

217.  It  is  evident  that  this  is  not  to  perform  addition,  but  only 
to  represent  it.  We  see  at  the  same  time,  however,  that  in 
order  to  perform  it  actually,  we  have  only  to  leave  out  the 
parentheses ;  for  as  the  number  d  -f  e  +/  is  to  be  added  to  the 
other,  we  know  that  this  is  done  by  joining  to  it  first  -f  rf, 
then  +  e,  and  then  +/;  which  therefore  gives  the  sum  a  -^-h  ■{- 

The  same  method  is  to  be  observed,  if  any  of  the  terms  are 
affected  with  the  sign  — ;  they  must  be  joined  in  the  same  way, 
by  means  of  their  proper  sign. 

218.  To  make  this  more  evident,  we  shall  consider  an  exam- 
ple in  pure  numbers.     It  is  proposed  to  add  the  expression  15 

—  6  to  12 —  8.     If  we  begin  by  adding  15,  we  shall  have  12 

—  8-1-15;  now  this  was  adding  too  much,  since  we  had  only 
to  add  15  —  6,  and  it  is  evident  that  6  is  the  number  which  we 
have  added  too  much.  Let  us  therefore  take  this  6  away  by 
writing  it  with  the  negative  sign,  and  we  shall  have  the  true 
sum,  12  —  8  -f  15  — -  6, 


60'  Mgehra.  Sect.  2. 

which  sliews  that  the  sums  are  found  by  writing  all  the  terms, 
each  with  its  proper  sign, 

219.  If  it  were  required  therefore  to  add  the  expression  d  —  e 
— /  to  c  —  6  +  c,  we  should  express  the  sum  thus  : 

a  —  6  +  c-frf  —  e  —  /, 
remarking  however  that  it  is  of  no  consequence  in  what  order 
we  write  these  terms.    Their  place  may  be  changed  at  pleasure, 
provided  tlieir  signs  be  preserved.    This  sum  might,  for  exam- 
ple, be  written  thus : 

c  —  e  +  a  — /  -f  rf  —  6. 

220.  It  frequently  happens,  that  the  sums  represented  in 
this  manner  may  be  considerably  abridged,  as  when  two  or 
more  terms  destroy  each  other  ;  for  example,  if  we  find  in  the 
same  sum  the  terms  -f-  a  —  a,  or  3a  —  4a  +  a  .•  or  when  two 
or  more  terms  may  be  reduced  to  one.  Examples  of  this  second 
reduction : 

3a  +  2a  =  5fl  ;  7b  —  Sb  =  +  4b   ; 

—  6c  4-  10c  =  +  4c ; 

5a  —  8a  =  —  3a  ;  —  7b  +  b=:  —  6&  ; 

—  5c  —  4c  =  —  7c 

2a  —  5a  -f-  a  =  —  2a  ;  —  S6  —  56  +  26  =  —  6&. 

Whenever  two  or  more  terms,  therefore,  are  entirely  the  same  with 

regard  to  letters,  their  sum  may  be  abridged :  but  those  cases 

must  not  be  confounded  with  such  as  these,  2aa  +  3a,  or  26^ 

^—  6*,  which  admit  of  no  abridgment. 

221.  Let  us  consider  some  more  examples  of  reduction  ;  the 
following  will  lead  us  immediately  to  an  important  truth.  Sup- 
pose it  were  required,  to  add  together  the  expressions  a  +  6  and 
a  —  b;  our  rule  gives  a  -{-b  -\-  a  —  b;  now  a  -f-  a  =  2a  and  b 
—  6  =  0;  the  sum  then  is  2a  .•  consequently  if  we  add  together 
the  sum  of  two  numbers  (a  4-  6)  and  their  difference  (a  —  6,) 
we  obtain  the  double  of  the  greater  of  those  two  numbers. 


Further  examples : 

3a  —  26  —  c 
56  —  6c  4-  a 


4a  +  36 —  7c 


a 3  —  QdQjf  4.  2abb 
—  aa6  4-  2a66  —  6* 

a3  —  3aa6  4. 4  a66  —  6^ 


Chap.  2.  Of  Compound  ^uantitie9,  ^1 

CHAPTER.  11. 

Of  the  Subtraction  of  Compound  Quantities. 

222.  If  we  wish  merely  to  represent  subtraction,  we  Inclose 
each  expression  within  two  parentheses,  joining,  by  the  sign  — , 
the  expression  which  is  to  be  subtracted  to  that  from  which  we 
have  to  subtract  it. 

When  we  subtract,  for  example,  the  expression  d  —  e  +/ 
from  the  expression  a  —  6  -f  c,  we  write  the  remainder  thus  : 

(-a  —  b  +  cj-^fd-^e  ^f)  ; 
and  this  method  of  representing  it  sufficiently  shews,  which  of 
the  two  expressions  is  to  be  subtracted  from  the  other. 

223.  But  if  we  wish  to  perform  the  actual  subtraction,  we 
must  observe,  first,  when  we  subtract  a  positive  quantity  -f  b 
from  another  quantity  a,  we  obtain  a  —  b  :  and  secondly,  when 
we  subtract  a  negative  quantity  —  b  from  a,  we  obtain  a  -\-b  ; 
because  to  free  a  person  from  a  debt  is  the  same  as  to  give  him 
something. 

224.  Suppose,  now,  it  were  required  to  subtract  the  expres- 
sion b  —  d  from  the  expression  a'-^Cy  we  first  take  away  b  ; 
which  gives  a  —  c  —  b.  Now  this  was  taking  too  much  away 
by  the  quantity  d,  since  we  had  to  subtract  only  b  —  d;  we  must 
therefore  restore  the  value  of  d,  and  shall  then  have 

a  —  c  —  b  -{•  d; 
whence  it  is  evident,  that  the  terms  of  the  expression  to  be  sub- 
tracted must  change  their  signs,  and  be  joined,  with  the  contrary 
signs,  to  the  terms  of  the  other  expression. 

225.  It  is  easy,  therefore,  by  means  of  this  rule,  to  perform 
subtraction,  since  we  have  only  to  write  the  expression  from 
which  we  are  to  subtract,  such  as  it  is,  and  join  the  other  to  it 
without  any  change  beside  that  of  the  signs.  Thus,  in  the  first 
example,  where  it  was  required  to  subtract  the  expression  d  —  e 
4-/  from  a  —  6  -f  c,  we  obtain  a  —  b  -\-  c  —  d-f-e  — /. 

An  example  in  numbers  will  render  this  still  more  clear.  If 
we  subtract  6  —  2  -|-  4  from  9  —  3  -f  2,  we  evidently  obtain 

9  —  3  -{-  2— •  6-f-2  — 4; 
for  9  —  3  -f 2  =  8  J  also,  6  —  2+4  =  85  now  8  —  8  :=  0. 


6S  Mgebra.  Sect.  2, 

226.  Subtraction  being  therefore  subject  to  no  difficulty,  we 
have  only  to  remark,  that,  if  there  are  found  in  the  remainder 
two,  or  more  terms  which  are  entirely  similar  with  regard  to 
the  letters,  that  remainder  may  be  reduced  to  an  abridged  form, 
by  the  same  rules  which  we  have  given  in  addition. 

227.  Suppose  we  have  to  subtract  from  a  -f  &,  or  from  the 
sum  of  two  quantities,  their  difference  a  —  6,  we  shall  then  have 
a  +  b  —  a  +  6 ;  now  a  —  a  =  0,  and  6  +  6  =  2&  ;  the  remain- 
der sought  is  therefore  2&,  that  is  to  say,  the  double  of  the  less 
of  the  two  quantities. 

228.  The  following  examples  will  supply  the  place  of  further 
illustrations : 


aa-^-ah  +  6& 
hb  -^ab  —  an 


3a^-4&  +  5c 
26  4. 4c  —  6a 


a^  -f-3ao&  -f  5abb  -f  b^ 
as  —  Saab  -f  Sabb  —  b^ 


V/a_-h2  V6_ 


2aa 


9a  —  6b  +  c. 


6aab  +  Qb^. 


+  5  V6 


CHAPTER  III. 

Of  the  Multiplication  of  Compound  ^antities. 

S29.  Whex  it  is  only  required  to  represent  multiplication, 
we  put  each  of  the  expressions,  that  are  to  be  multiplied  together, 
.within  two  parentheses,  and  join  them  to  each  other,  sometimes 
without  any  sign,  and  sometimes  placing  the  sign  x  between 
them.  For  example,  to  represent  the  product  of  the  two  expres- 
sions a  —  b  +  c  and  d  —  e  -|-  /,  when  multiplied  together,  we 
write 

(^a^b  +  cjx  Cd  —  e^f.J 

This  method  of  expressing  products  is  much  used,  because  it 
immediately  shews  the  factors  of  which  they  are  composed. 

230.  But  to  shew  how  a  multiplication  is  to  be  actually  per- 
formed, we  may  remark,  in  the  first  place,  that  in  order  to 
multiply,  for  example,  a  quantity,  such  as  a  —  b  -^  c,  by  2, 
each  term  of  it  is  sepai^ately  multiplied  by  that  number ;  so  that 
the  product  is 

2a  —  2&  -f.  2c. 


Ckap.  3.  Of  Compound  ^lantities.  68 

Now  the  same  thing  takes  place  with  regard  to  all  other 
numbers.     If  d  were  the  number,  by  which  it  was  required  to 
multiply  the  same  expression,  we  should  obtain 
ad  —  hd  ■{■  cd, 

231.  We  just  now  supposed  that  d  was  a  positive  number ;  but 
if  the  factor  were  a  negative  number,  as  —  e,  the  rule  formerly 
given  must  be  applied ;  namely,  that  two  contrary  sigiis  multi- 
plied  together  produce  — ,  mid  that  two  like  signSi  give  +. 

We  shall  then  have ; 

—  ae  -i-be  —  ce. 

232.  To  shew  how  a  quantity,  A,  is  to  be  multiplied  by  a 
compound  quantity,  d  —  e  ;  let  us  first  consider  an  example  in 
common  numbers,  supposing  that  A  is  to  be  multiplied  by  7  —  3. 
Now  it  is  evident,  that  we  are  here  required  to  take  the  quad- 
ruple of  A :  for  if  we  first  take  A  seven  times,  it  will  then  be 
necessary  to  subtract  3A  from  that  product. 

In  general,  therefore,  if  it  be  required  to  multiply  by  d  —  e, 
we  multiply  the  quantity  A  first  by  d  and  then  by  e,  and  sub- 
tract this  last  product  from  the  first :  whence  results  dA  —  eA. 
Suppose  now  A  =  a  —  &,  and  that  this  is  the  quantity  to  be 
multiplied  by  d  —  e;  we  shall  have 

dA  =:.  ad  —  hd 
eA  =  ae  —  be 


whence  the  product  required  =  ad  —  bd  —  ae  +  be, 

233.  Since  we  know  therefore  the  product  (a —  6)  X  (cd  —  e,) 
and  cannot  doubt  of  its  accuracy,  we  shall  exhibit  the  same 
example  of  multiplication  under  the  following  form  : 

a  —  b 

d  —  e 


ad  —  bd  —  ae  +  be. 
This  shews,  that  we  must  multiply  each  term  of  the  upper  ex- 
pression by  each  term  of  the  lower^  and  that,  with  regard  to  the 
signs,  we  must  strictly  observe  the  rule  before  given ;  a  rulo^ 
which  this  would  completely  confirm,  if  it  admitted  of  the  least 
doubt. 


(34  Mgehra,  Sect.  2. 

234.  It  will  be  easy,  according  to  this  rule,  to  perform  the 
following  example,  which  is,  to  multiply  a  +  ft  by  a  —  b: 

a+& 
a  —  b 


aa  -f  ab 
—  ab  —  bb 

Product    aa  —  bb. 

235.  Now  we  may  substitute,  for  a  and  b,  any  determinate 
numbers ;  so  that  the  above  example  will  furnish  the  following 
theorem  ;  viz.  The  product  of  the  sum  of  two  numbers,  multU 
plied  by  their  difference,  is  equal  to  the  difference  of  the  squares  of 
those  numbers.    This  theorem  may  be  expressed  thus  : 

(a  +  6)  X  (a  —  b)=zaa  —  bb. 
And  from  this,  another  theorem  may  be  derived ;  namely. 
The  difference  of  two  square  numbers  is  alwaijs  a  product,  and 
divisible  both  by  the  sum  and  by  the  difference  of  the  roots  of  those 
two  squares  ;  consequently  the  difference  of  two  squares  can  never 
be  a  prime  number, 

236.  Let  us  now  perform  some  other  examples  : 

I.)  2a  —  3 
a  +2 


2aa  —  5a 
+  4a 


Qaa  +  a  —  6. 


II.)  4aa  —  6a  4-  9 

2a  +  3 


8a*  —  12aa  +  18a 

4-  12aa —  18a +  27 


8a3  4-27 


Chap.  3.  Of  Compound  Quantities^,  65 

III.)  3aa  —  2ab  —  bb 

2a  — 4b 


6a  3  —  4aab  —  2abb 

—  I2aab  +  Sabb  +  46» 

6fl3  —  iQaaf,  ^  Qabb  +  46«. 


IV.)  aa  +  2ab  +  Qbb 

aa —  2ab  +  2bb 

a*  4-  Sa^ft  +  2aabb 

—  2a*6  —  4aabb  — 4ab^ 

4-  2fla66  -f-  4ab^  +  4&* 

a*  +  464 


V.)  2aa  —  3a6  —  4bb 
3aa  —  2«6  +  bb 


6a^  —  9a3  —  12aabb 

—  4aS6  +  6aabb  +  Safe^ 

+  2aa66  —  Sab^  —  46'* 

6a4  —  13^3^  _  4aabb  +  5a6«  —  46* 


VI.)  aa  -{-bb  -}.  cc  —  ab  —  ac  —  be 
a  +    6  +  c 

a^  -^  abb  -\-  ace  —  aab  —  aac  —  abc 

aab  +  6^    ^  bcc  —  abb  —  abc  —  bbc 

aac  -f-  bbc  +  c*    —  abc —  ace  —  bee 

a^  —  3a6c  +  6^  +  c^. 

237.  When  we  have  more  than  two  quantities  to  multiplij  to- 
gether, it  xviU  easily  be  understood  that,  after  having  multiplied 
two  of  them  together ,  we  must  then  multiply  that  product  by  one 
of  those  which  remain,  and  so  on.  It  is  indifferent  what  order  is 
observed  in  those  multiplications, 
9 


66  Mgebra.  Sect.  2. 

Let  it  be  proposed,  for  example,  to  find  the  value,  or  product, 
of  the  four  following  factors,  x'iz;. 

L  IL  III.  IV. 

(a  4-  6)  (aa  +  aft  +  hh)  (a  —  b)  (aa  —  ab  -{-  56.) 
We  will  first  multiply  the  factors  I.  and  II. 
Ih  aa  ^ab  +  bb 
I.      a+    b 


a^  +  aab  -f  abb 
-{- aab  +  abb -^  ¥ 


I.  II.  a^  +  Qaab  +  2abb  -f-  fe^. 
Next  let  us  multiply  the  factors  III.  and  IV. 
IV.  aa  —  a6  +  bb 
III.    a  — & 


O^  —  a^b  -f-  abb 
—  a^b  +  abb  —  6* 

III.  IV.  a^  --  2aab  +  Qabb  —  bK 
It  remains  now  to  multiply  the  first  product  I.  II.  by  this 
second  product  III.  IV  : 
a^  +  2aab  +  2abb  -f  6*     I.  II. 
as  —  2aab  +  ^abb  —  b^     III.  IV. 


a«  +  2o«6  +  2a*&6  +  a^b^ 

—  2a«6  — 4a*6&  —  4a3&'  —  2fla6* 

2a^bb  +  4a363  -f  4afl6'*  +  Qab^ 

—  a^b^  —  2aab*  —  2fl6«  —  b^ 

a^  -—-b^ 
And  this  is  the  product  required. 

238.  Let  us  resume  the  same  example,  but  change  the  order 
of  it,  first  multiplying  the  fractions  I.  and  III.  and  then  II.  and 
IV.  together. 

I.  a  -f-  & 
III.  a  —  6 


aa  4-  ab 
^ab^bb 


I.  III.  z=:aa  —  bb. 


Chap.  3,  OJ  Compmnd  ^antities,  6f 

II.  aa  •}-  ab  +  bb 
IV.  aa  —  ab  -^bb 


a*  +  a^b  4-  aabb 

—  a^lf  —  aabb  —  ab^ 

aahb  +  ab^  +  ft* 

II.  IV.  =  a*  -f-  aabb  -f  6* 
Then  multiplying  the  two  products  I.  III.  and  II.  IV. 
II.  IV.  =  a*  -f-  aabb  +  6* 
I.  III.  =zaa-^bb 


a*  4-  aHb  +  aab^ 
—  a*bb  —  aab* 


We  have  a*  —  6«, 
which  is  the  product  required. 

239.  We  shall  perforin  this  calculation  in  a  still  different 
manner,  first  multiplying  the  P.  factor  hy  the  IV*^.  and  next 
the  II**.  by  the  lU^. 

IV.  aa  —  db   -f  bb 
I.       a  -\-  b 


a^  —  aab  -f-  abb 

abb — abb  -^b^ 


I.  IV.  =  a»  +  fe- 


II. aa  +  aft  -f-  bb 

III.  a  — & 


a'  -I-  aab  +  abb 
—  aab --^  abb —  6* 

II.  III.  =  a^— feT 

It  remains  to  multiply  the  product  I.  IV.  and  II.  III. 
I.  IV.  =a3  +63 
II.  III.  =a3  — 6' 


a«  +  a^b^ 
and  we  stiU  obtain  a*  —  6«  as  before. 


f»  *  Mgebra,  Sect,  2. 

240.  It  will  be  proper  to  illustrate  this  example  by  a  numeri- 
cal application.  Let  us  make  a  =  3  and  &  =  2,  we  shall  have 
a  -f  &  =  5  and  a  —  6  =  1  ;  further,  aa  =  9,  ab  =  6,  6&=4. 
Therefore  aa  -f  oft  -f  56  =  19,  and  aa  —  ah  -}.bbz=  7.  So  that 
the  product  required  is  that  of  5x19x1x7,  which  is  665. 

Now  a 6  =  729,  add  6«  =  64,  consequently  the  product  re» 
quired  is  a®  —  b^  =  665,  as  we  have  already  seen. 


CHAPTER  IV. 

Cff  the  Division  of  Compound  Quantities* 

141.  When  we  wish  simply  to  represent  division,  we  make 
use  of  the  usual  mark  of  fractions,  which  is,  to  write  the  de- 
nominator under  the  numerator,  sepai^ating  them  by  a  line ;  or 
to  inclose  each  quantity  between  parentheses,  placing  two  points 
between  the  divisor  and  dividend.  If  it  w^ere  required,  for 
example,  to  divide  a  -f-  6  by  c  -f-  d  we  sliould  represent  the  quo- 
tient thus  ,  according  to  the  former  method ;  and  thus, 
c  -y-  d 

(a  -f  6) :  (c  +  d)  according  to  the  latter.     Each  expression  is 
read  a  +  6  divided  by  c  +  d. 

242.  When  it  is  required  to  divide  a  compound  quantity  by  a 
simple  one,  we  divide  each  term  separately.  For  example ;  6a  —  Sb 
-f  4c  divided  by  2,  gives  3a  —  46  -f  2c  ;  and  (^aa  —  2a6)  :  (a) 
=  a  —  26.  In  the  same  manner,  (a^  —  2afl6  -f  3aa6)  :  (a)  = 
aa  —  2a6  -f  366 ;  (4aa6  —  6aac  ■+-  8a6c)  :  (ea)  =  2a6  —  3ac  -f- 
46c  ;  (9aabc  —  12ahhc  -f  15a6cc)  :  (3a6c)  =  3a  —  46  -f  5c,  &c. 

243.  If  it  should  happen  that  a  term  of  the  dividend  is  not 
divisible  by  the  divisor,  the  quotient  is  represented  by  a  fraction, 

as  in  the  division  of  a  -f  6  by  a,  which  gives  1  -^ — .    Likewise, 

(aa  —  a6  +  66)  :  (aa)  =  1  —  -  -f-  - . 
For  the  same  reason,  if  we  divide  2a  -f  6  by  2,  we  obtain 
a  +  •-  ;  and  here  it  may  be  remarked,  that  we  may  write  —6, 


Chap.  4.  Of  Compound  ^uintities,  69 

instead  of  — ,  because—  times  b  is  equal  to  — .     In  the    same 

b  1  2b  2 

manner  —  is  the  same  as  —b,  and  —  the  same  as  —6,  &c. 

244.  But  when  the  divisor  is  itself  a  compound  quantity, 
division  becomes  more  difficult.  Sometimes  it  occurs  where  we 
least  expect  it ;  but  when  it  cannot  be  performed,  we  must  con- 
tent ourselves  with  representing  the  quotient  by  a  fraction,  in 
the  manner  that  we  have  already  described.  Let  us  begin  by 
considering  some  cases,  in  which  actual  division  succeeds. 

245.  Suppose  it  were  required  to  divide  the  dividend  ac  —  be 
by  the  divisor  a  —  &,  the  quotient  must  then  be  such  as,  when 
multiplied  by  the  divisor  a  —  b,  will  produce  the  dividend  ac  —  be. 
Now  it  is  evident,  that  this  quotient  must  include  c,  since  with- 
out it  we  could  not  obtain  ac.  In  order,  therefore,  to  try 
whether  c  is  the  whole  quotient,  we  have  only  to  multiply  it  by 
the  divisor,  and  see  if  tliat  multiplication  produces  the  whole 
dividend,  or  only  part  of  it.  In  the  present  case,  if  we  multiply 
a  —  b  by  c,  we  have  ac  —  6c,  which  is  exactly  the  dividend  ; 
so  that  c  is  the  whole  quotient.     It  is  no  less  evident,  that 

(aa  -^  ab)  :  (a  +  &)  =  a  ;  (3aa  —  Qab)  :  (3a  —  26)  =  a ;  (6aa 
—  9ab)  :  (2a  —  36)  =  Sa,  &c. 

246.  TVe  cannot  failf  in  this  way,  to  find  apart  of  the  quotient ; 
if  therefore,  what  we  have  found,  when  multiplied  by  the  divisor, 
does  not  yet  exhaust  the  dividend,  we  have  only  to  divide  the 
remainder  again  by  the  divisor,  in  order  to  obtain  a  second  part  of 
the  quotient ;  and  to  continue  the  same  method,  until  we  have  found 
the  whole  quotient. 

Let  us,  as  an  example,  divide  aa  -f  3a6  +  266  by  a  +  6  ;  it  is 
evident,  in  the  first  place,  that  the  quotient  will  include  the  term 
o,  since  otherwiae  we  should  not  obtain  aa»  Now,  from  the 
multiplication  of  the  divisor  a  -f  6  by  a,  arises  aa  -j-  ah  ;  whicli 
quantity  being  subtracted  from  the  dividend,  leaves  a  remainder 
2a6  4-  266.  This  remainder  must  also  be  divided  by  a  +  6  ;  and 
it  is  evident  that  the  quotient  of  this  division  must  contain  the 
term  26.  Now  26,  multiplied  by  a  -f  6,  produces  exactly  2a6  -|- 
266;  consequently  a  -|-  26  is  the  quotient  required  ,•  whicij,  mul- 


ro  ^  Algebra.  Sect.  2. 

tiplied  by  the  divisor  a-\-h,  ought  to  produce  the  dividend  aa  -f- 
Sab  +  2&&.     See  the  whole  operation  : 

a  '\-b^  aa-\-  Sab  -|-  9,bb  (a  +  2& 


2a6  +  266 
2a&  +  266 

0. 

247.  TMs  operation  will  be  facilitated  if  we  choose  one  of  the 
terms  of  the  divisor  to  be  written  first,  and  then,  in  arranging  the 
terms  of  the  dividend,  begin  with  the  highest  powers  of  that  first 
term  of  the  divisor.  This  term  in  the  preceding  example  was  o; 
the  following  examples  will  render  the.  operation  more  clear. 

a  —  6)  a3  —  Saab  4-  Sabb  -—  6*  {m  —  2a6  +  66 

ft3   (ifiJj 


2aab  +  3a66 
2aa6  -f  2a66 


a66  —  b^ 
abb  —  6» 


a  -f  6)  oa  —  66  (a  -— 
aa  +  ab 


—  a6- 

—  a6^ 

-66 
-66 

0. 

—  26) 

18aa  — 
18aa  — 

-  866  (6a 
-Uab 

+  46 

12fl6  — 
12a6-- 

■  866 

■  866 

3fl 


p 

Chap.  4.  Of  Compound  ^antities,                            7t 

a  +  b)  a^  +  6*  (aa  —  ab  +  66 

a^  -f  aab 


—  006  -f  6* 
--.  aa6  —  a66 

a66  +  68 
abb  +  b^ 

0. 

2a- 

-6)  8a»  —  6'  (4aa  +  2a6  +  66 
8a3  —  4aab 

4aab  —  6^ 
4aab  —  2a66 

2a66  —  63 
2a66  —  63 

aa  —  2a6  -j-  66)  a*  —  4a36  +  6aa66  —  4a63  4. 6* 
aa  —  2a6  +  66)  a*  —  20^6  +  aa66 

—  2a36  +  5aa66  — 4a63 

—  2a36  +  4aa66  —  2a63 

fla66  —  2a63  +  b* 
aabb  —  206^  4.  b* 

aa  —  2a6  +  466)  a-*  +  4aa66  +  166*  (^aa  —  2a6  +  466 
a*  —  2a3  6  -|-4aa66 


2a86  +  166* 
2a^b'^4aabb  +  8a63 

4aa66  — 8a63-f-  166* 
4aa66  —  8a63  +  166* 


^2  Mgebra.  Sect.  2. 

aa  —  2a&  -f  266)  fl*  +  46*  {aa  +  2r/6  +  266 


2a  3  5  —  2aabb-{.4b^ 
^a^h  —  4aa66  +  4^6^ 


2aa66  —  4a63  +  46* 
2aa66  —  4a63  ^  46* 


0. 


1  —  2a;  +  xx)  1  —  5x  -}-  \Oxx  —  IQx^  +  5x^  —  x^ 
1  —  So?  4-  3^x  —  a?3)  1  —  9,x  -^  XX 


—  3x  +  9xx  —  lOx^ 


3x0?  —  7x^  +  5o?* 
3a?a?  —  6^3  +  So;* 

—  x^  +  2x*  —  X* 

—  x3  +  2x*  —  X* 


CHAPTER  V. 

Of  the  Resolution  of  Fractions  into  infinite  Series, 

248.  When  the  dividend  is  not  divisible  by  the  divisor,  the 
quotient  is  expressed,  as  we  have  already  observed,  by  a  frac- 
tion. 

Thus,  if  we  have  to  divide  1  by  1  —  a,  we  obtain  the  fraction 

— — .    This,  however,  does  not  prevent  us  from  attempting  the 

division,  according  to  the  rules  that  have  been  given,  and  con- 
tinuing it  as  far  as  we  please.  We  shall  not  fail  to  find  the  true 
quotient,  though  under  different  forms. 


Chap.  5.  Of  Compound  Quantities.  75 

249.  To  prove  this,  let  us  actually  divide  the  dividend  1  by 
the  divisor  1  —  a,  thus : 

1— «)1     (1+^;  or,l— a)l     (i  +  a+j-^ 
l—-a  1  —  a 


remainder  a  a 

a  — aa 


remainder  aa 
To  find  a  greater  number  of  forms,  we  have  only  to  continue 
dividing  aa  by  1  —  a  ; 


a  3  /,4 


1  — a)  aa    (aa  + ,  then  1  —  a)    a^  (a^  + 

aa — a^  a^  —  a* 


and  again  1  —  a)  a*      {a*  + 


a*,  & 
e 
under  all  the  following  forms 


£50.  This  shews  that  the  fraction  may  be  exhibited 

1  —  a 


I-)l+r^;    ll.)l+a  +  - 


aa 
f 


III.)  1  +  o  -f.  aa  +  -^ ;  IV.)  1  +  a  +  aa  +  aS  +  -^ ; 
1  — a  '       '        '  1  —  a 

V.)  1  +  a  +  aa  +  a3  -f  a*  +  -^^—,  &c. 

Now,  by  considering  the  first  of  these  exjjressions,  which  is 
a 


a  1    __  rt 

1  +  YZIT,^  ^"^  remembering  that  1  is  the  same  as  :; -,   we 

have 


1    ,       a           1  —  a          a           1 — a  4- a 
1  -f -= -  + — L„ 


1  —  a       1  —  a      1  —  a  1  —  a  1  —  a* 

If  we  follow  the  same  process  with  regard  to  the  second 

expression  1  +  a  +  j-^,  that  is  to  say,  if  we  reduce  the  in- 

10 


74  Jlgehra.  Sect.  2. 

teger  part  1  +  a  to  the  same  denominator  1  —  a  we  shall  have 

"~"^,  to  which  if  we  add  +  ,  we  shall  have  -IZf^Xf? 

1  —  a  1  —  a  1  —  a 

that  is  to  say, . 

1  — n 

In  the  third  expression,  1  -|-  a  +  aa  -{- ,  the  integers 

I    , ^3 

reduced  to  the  denominator  1  —  a  make  — ;  and  if  we 

1  —  a 

add  to  that  the  fraction ,  we  have ;  wherefore  all 

1  —  a  1  —  a 

these  expressions  are  equal  in  value  to ,  the  proposed 

Fraction. 

251.  This  being  the  case,  we  may  continue, the  series  as  far 
as  we  please,  without  being  under  the  necessity  of  performing 
any  more  calculations.    We  shall  therefore  have 

=  1  +  a  +  aa  +  a'  +  a*  +  a*  +  a«  -f  a^  -f  -^ —  ; 

1  — lO  1  —  a 

or  we  might  continue  this  further,  and  still  go  on  without  end. 
For  this  reason,  it  may  be  said,  that  the  proposed  fraction  has 
been  resolved  into  an  infinite  series,  which  isl  +a-faa-fa' 
^.a*  -f  a«  +a«  -{.  a"^  +  a^  +  a®  +ai«  +ai*  +a*^5  &c.  to 
infinity.  And  there  are  sufficient  grounds  to  maintain,  that  the 
value  of  this  infinite  series  is  the  same  as  that  of  the  fraction 
t 

252,  What  we  have  said  may,  at  first,  appear  surprising ; 
but  the  consideration  of  some  particular  cases  will  make  it  easily 
understood. 

Let  us  suppose,  in  the  first  place,  a=  1 ;  our  series  will 

become  1-fl  +  l  +  l  +  l  +  l  +  l,  &c.     The  fraction , 

to  which  it  must  be  equal,  becomes  — .  Now,  we  before  remark- 
ed, that  ~  is  a  number  infinitely  great ;  which  is,  therefore, 
here  confirmed  in  a  satisfactory  manner. 


i 


that 


Chap.  5.  Of  Compound  ^antities.  75 

But  if  we  suppose  a  =  2,  our  series  becomes  =1+2+4 

+  8  +  16  +  32  +  64,  &c.  to  infinity,  and  its  value  must  be -, 

is  to  say  =  —  1 ;  which  at  first  sight  will  appear  ab- 
surd. But  it  must  be  remarked,  that  if  we  wish  to  stop  at  any 
term  of  the  above  series,  we  cannot  do  so  without  joining  the 
fraction  which  remains.  Suppose,  for  example,  we  were  to  stop 
at  64,  after  having  written  i  +  2  +  4  +  8  +  16  +  32  +  64,  we 

128            128 
must  join  the  fraction -,  or ,   or  —   128  ;    we  shall 

1  "~~  Z>  "~~"  1 

therefore  have  127  —  128,  that  is  in  fact  —  1. 

Were  we  to  continue  the  series  without  intermission,  the  frac- 
tion indeed  would  be  no  longer  considered,  but  then  the  series 
would  still  go  on. 

253.  These  are  the  considerations  which  are  necessary,  when 
we  assume  for  a  numbers  greater  than  unity.  But  if  we  sup- 
pose a  less  than  1,  the  whole  becomes  more  intelligible. 

For  example,   let   a  =  ^  ;  we  shall  have  = 7  = 

*  *  '  1  — a      i  —  J 

—  =  2,  which  will  be  equal  to  the  following  series  :  1+^+1 

+ 1  +  tV  +  7f  +  A  +  T¥T»  ^^-  to  infinity.    Now,  if  we  take 
only  two  terms  of  this  series,  we  have  1  + 1,  and  it  wants  |, 

that  it  may  be  equal  to =  2.     If  we  take  three  terms,  it 

1  —  a 

wants  1 ;  for  the  sum  is  1|.     If  we  take  four  terms  we  have 

1|,  and  the  deficiency  is  only  -J.     We  see,  therefore,  that  the 

more  terms  we  take,  the  less  the  difference  becomes ,  and  that, 

consequently,  if  we  continue  on  to  infinity,  there  will  be  no 

difference  at  all  between  the  sum  of  the  series  and  2,  the  value 

of  the  fraction 


1  —a 


254.  Let  a=zl;   our  fraction will  be  = 


^,  ^_^  j_. 


s 


=  1|,  which,  reduced  to  an  infinite  series,  becomes  1  +  ^  +  ». 

+  ^V  +  TT  +  ii^9  &c.  and  to  which is  consequently 

equal. 


76  Algebra.  Sect.  2. 

When  we  take  two  terms,  we  have  1|,  and  there  wants  |.  If 
we  take  three  terms,  we  have  1^,  and  there  will  still  he  wanting 
y\.  Take  four  terms,  we  shall  have  1|4,  and  the  difference  is 
■j?^.  Since  the  error,  therefore,  always  becomes  three  times  less, 
it  must  evidently  vanish  at  last. 

1  1 

255.  Suppose  a  =  4  ;    we  shall  have = r-=  3,  and 

^  1—  a       1  — I 

the  series  1  + 1  +  |  +  ^s^  -f.  |6  4.  ^^^,  &c.  to  infinity.  Taking 
first  1-|,  the  error  is  1-|  ^  taking  three  terms,  which  make  2j, 
the  error  is  | ;  taking  four  terms  we  have  2|4^,  and  the  error 

ic    16 
^  IT* 

256.  If  a  =  1,  the  fraction  is j-  =  —  =  1^  ;  and  the  se- 

ries  becomes  1  +  i  +  tV  +  tV  +  ^tt>  ^c*  '^^^  *wo  first  terms, 
making  1  +  ^,  will  give  y\  for  the  error ;  and  taking  one  term 
more,  we  have  IJ-g,  that  is  to  say,  only  an  error  of  ■^■^, 

257.  In  the  same  manner,  we  may  resolve  the  fraction , 

into  an  infinite  series  by  actually  dividing  the  numerator  1  by 
the  denominator  1  -f  a,  as  follows  : 

1  +  a)    1    (1  —  a  ^aa  —  a^  -{•  a^ 
l+a 


f 


a 

a  —  da 


aa 

aa  +  a^ 


—  a*,  &c. 
Whence  it  follows,  that   the  fraction   t-jT^  ^^  ®^"^^  *^  ^^^* 

series, 

1  —  a  +  aa  —  a3  +  a*  —  a«  +  a«  —  a%  &c. 


Chap.  5*  Of  Compound  Quantities.  77 

258,  If  we  make  o  =  1,  we  have  this  remarkahle  comparison  : 
— i—  —  1  =  1  —  1  +  1  —  1  -f  1  ^  1  +  1  —  1,  &c.  to  infinity. 


1  -f  a 

This  will  appear  rather  contradictory  ;  for  if  we  stop  at  —  1, 
the  series  give  0 ;  and  if  we  finish  by  +  1,  it  gives  1.  But  this 
is  precisely  what  solves  the  difficulty  ;  for  since  we  must  go  on 
to  infinity  without  stopping  either  at  —  1,  or  at  -f  1,  it  is  evi- 
dent, that  the  sum  can  neither  be  0  nor  1,  but  that  this  result 
must  lie  between  these  two,  and  therefore  be  =  |. 

259.  Let  us  now  make  a  =  i,  and  our  fraction  will  be 


" 1+^ 

=  |,  which  must  therefore  express  the  value  of  the  series  1  —  J 
+  i  —  T  +  tV  —  A  +  A'  ^c*  *^  infinity.  If  we  take  only 
the  two  leading  terms  of  this  series,  we  have  |-,  which  is  too 
small  by  \.  If  we  take  three  terms,  we  have  |,  which  is  too 
much  by  -^^.  If  we  take  four  terms,  we  have  4,  which  is  too 
small  by  -^^^  &c. 

260.  Suppose  again   fl  =  -J  ?  ^^^  fraction  will  be  = 

=  |,  and  to  this  the  series  1  —-  4  -f  |.  —  -\.  +  „i_.  —  ^1^  + 
y|^,  &c.  continued  to  infinity,  must  be  equal.  Now,  by  con- 
sidering only  t\NO  terms,  we  havef,  which  is  too  small  by  -j?^. 
Three  terms  make  -J,  which  is  too  much  by  ^^,  Four  terms 
make  |^,  which  is  too  small  by  ^J^,  and  so  on. 

261.  The  fraction  -— ; —  may  also  be  resolved  into  an  infinite 

1  +  a       '' 

series  another  way ;  namely,  by  dividing  1  by  a  -f  1,  as  follows : 


78  Mgebra.  Sect.  2. 

,1         11         1.1 

^+4 


1 

a 

a       oa 


J. 

aa 


i  +  -i 


aa       a" 


JL 

a"' 

1  —  i 

a3       a* 


_i^,  &c. 

Consequently,  our  fraction  ,  is   equal  to  the  infinite 

series  i  —  I    +  i  —  -1   +  i  —  i,  &c.    Let  us  make 
a         aa  ^   a^         a^         a*  a^ 

a  =  1,  and  we  shall  have  the  series  1  —  1  +  1  —  1+1  —  1> 
&c.  =  |,  as  before.  And  if  we  suppose  a  =  2,  we  shall  have 
the  series  J  —  J  +  |  —  ^V  +  A  —  A»  &c.  =  f 

262.  In  the  same  manner,  by  resolving  the  general  fraction 

—^—r  into  an  infinite  series,  we  shall  have, 
a  +  0 


Chap.  5.  OJ  Compound  ^antities.  79 


,v         ^  Q       be       bbe        6'c 
•^    ^         ^  a      aa       a^         a^ 


^ 

frc 

a 

be        bbc 

a         aa 

bbc 

• 

aa 

bbc 

bbbe 
+  «3 

aa 

63c 

' 

a3 

fcsc 

b*c 

a* 

a^ 

b^c 

a^   ' 

t  appears,  that 

we  may 

compare 

'  c 
r  with  the  series 

a+  6 

C  be       ,       66c  63c       c  .       '     n     'J. 

7-  —  ;;:;  +  -TX  —  TT-^  &c,  to  infinity. 


ft  aa  tt^  a 


Let  a  =  2,  6  =  4,c  =  S,  and  we  shall  have 


a  4-  6      4-1-2 
=  I  =  I  =  I  —  3  +  6  —  12,  &c. 

Let  a  =  10,  6  =  1  and  c  =  11,  and  we  have 

a  +  6 

^1    —    11   11-L_11        11  RfC 

—  ^  —  TTT  TOTT  T^  To?)  (T  TTJ^TJlTo'  "''^* 

If  we  consider  only  one  term  of  this  series,  we  have  W^  which 
is  too  much  by  -^^  ;  if  we  take  two  terms,  we  have  ^y^,  which 
is  two  small  by  -^^^ ;  if  we  take  three  terms,  we  have  -r?^J> 
which  is  two  much  by  ^^V^'  ^^' 

263.  When  there  are  more  than  two  terms  in  the  divisor,  we 
may  also  continue  the  division  to  infinity  in  the  same  manner. 

Thus,  if  the  fraction were  proposed,  the  infinite 

1  —  a  +  aa  ^    ^ 

series,  to  which  it  is  equal,  would  be  found  as  follows : 


80  Mgehra.  Sect.  2. 

1  —  a  +  fl«)  I  (1  +  a  ^  a^  —  a*  +  a6  -f  a',  &c. 

1  —  a  +  aa 


a  —  aa 

a  —  aa  +  a^ 


— -  a» 

—  a*  +  a*  —  a^ 

—  a*  4-  a* 

—  a^  +  a*  —  a^ 


a'' 

a6  ^  a?  +  «» 


a?  —  a8  4-  a^ 


We  have  therefore  the  equation  of  =  1  +  a  —  a^ 

1  •—  a  +  «« 

—  a-*  +  a^  +  a^  —  a^  —  a^<>,  &c.  Here,  if  we  make  a  =  1, 
we  have  1  =1+1—  1  —  1+1  +  1  —  1  —  1  +  1  +  1,  &c. 
which  series  contains  twice  the  series  found  above  1  —  1+1 

—  1+1,  &c.  Now,  as  we  have  found  this  =4?  it  is  not 
astonishing  that  we  should  find  |,  or  1,  for  the  value  of  that 
which  we  have  just  determined. 

Make  a  =  |j  and  we  shall  then  have  the  equation  —  =4  =  1 
fi  1 1-Li4._i     1       Ik'r 

T-  ■?  ¥  T 6-  T^  •g-4   T^  T"?"?  TT-J'  *^'^' 

Suppose  a  =  A,  we  shall  have  the  equation  ~  =  |  =  l+4  — 

■5 
A  —  TT  +  rlg^j  ^c.    If  we  take  the  four  leading  terms  of  this 
series,  we  have  VV>  which  is  only  y|y,  less  than  ^. 

Suppose  again  a  =  f?  we  shall  have  —  =  |  =  1  +  |  —  ■^\ 

-9 
— 14  +  tVt*  ^c*  this  series  must  therefore  be  equal  to  the  pre- 
ceding one  ;  and  subtracting  one  from  the  other,  -J  —  /y  —  14  + 
y«,*^,  must  be  =  0.  These  four  terms  added  together  make  — /-j. 

264.  The  method,  which  we  have  explained,  serves  to  resolve, 
generally,  all  fractions  into  infinite  series  5  and,  therefore,  it  is 


Chap.  6.  Of  Comjmund  ((uantities.  3i 

often  found  to  be  of  the  greatest  utility.  Furtlicr,  it  is  remark- 
able, that  an  infinite  series,  thoiigh  it  never  ceases/may  have  a 
determinate  value.  It  may  be  added,  tliat  from  this  branch  of 
mathematics  inventions  of  the  utmost  importance  have  been 
derived,  on  wliich  account  the  subject  deserves  to  be  studied 
with  the  greatest  attention. 


CHAPTER  VI. 

Of  the  Squares  of  Compound  Quantities, 

£65.  When  it  is  required  to  find  the  square  of  a  compound 
quantity,  we  have  only  to  multiply  it  by  itself,  and  the  product 
will  be  the  sqnare  required. 

For  example,  the  square  of  a  -^-h  is  found  in  the  following 
manner : 

a  ■\-b 
a-\-h 


aa  +  a6 

ab  +  hb 

aa  -f  2ab  -f-  bb, 

266.  So  that,  when  the  root  consists  of  two  terms  added  together^ 
as  a  -t-  bf  the  square  comprehends,  1st  tJie  square  of  each  term, 
namely  aa  and  bb ;  Qdly  twice  the  product  of  the  two  terms, 
namely  2afe.  So  that  the  sum  aa  -f-  2ab  -f  bb  is  the  square  of 
a  -\-b.  Let,  for  example,  a  =  10  and  6=3,  that  is  to  say,  let 
it  be  required  to  find  the  square  of  13,  we  shall  have  100  +  60 
+  9,  or  169. 

267.  We  may  easily  find,  by  means  of  this  formula,  the 
squares  of  numbers,  however  great,  if  we  divide  them  into  two 
parts.  To  find,  for  example,  the  square  of  57,  we  consider  that 
this  number  is  =  50  -f  7  ;  whence  we  conclude  that  its  square 
is  =  2=^00  +  700  +49  =  3249. 

268.  Hence  it  is  evident,  that  the  square  of  a  +  1  will  be  aa 
+  2rt  +  1  :  now  since  the  square  of  a  is  aa,  we  find  the  square 

11 


^t  Msehra.  Sect.  2. 


"£> 


a  -f  1  by  ailding  to  tliat  2a  +  1 ;  and  it  must  be  observed,  that 
this  2rt  4- 1  is  the  sum  of  the  two  roots  a  and  a  +  1. 

Thus,  as  the  square  of  10  is  100,  that  of  11  will  be  100  -f  21. 
The  square  of  57  being  3249,  that  of  58  is  3249  +  115  =  3364. 
The  square  of  59  =  3364  +  117  =  3481  ;  the  square  of  60  = 
3481  +  119  =  3600,  &c. 

269.  Tlie  square  of  a  compound  quantity,  as  a  -f  &,  is  repre- 
sented in  tins  manner :  (a  -f.  6)2.  We  have  then  (a  -f-  H)^  =  aa 
+  9.ab  -f  hh,  whence  we  deduce  the  following  equations  : 

(a  +  1)'  =  fla  +  2a  -f  1 ;  (a  +  2)3  =  aa  +  4a  +  4 ; 
(a  -f-  3)«  =  aa  -f-  6a  -f-  9 ;  (a  +  4)«  =  aa  -i~  8a  +  16  ;  Sec. 

270.  Tf  the  root  is  a  —  b,  the  square  of  it  is  aa  —  2ab  +  bb, 
which  contains  also  the  squares  of  the  two  terms,  but  in  such  a 
manner  that  ive  must  take  from  their  sum  twice  the  product  of 
tliose  two  terms. 

Let,  for  example,  a  =  10  and  6  =  —  1,  the  square  of  9  will 
be  found  =  100  —  20  +  1  =  81. 

271.  Since  wc  have  the  equation  (a  —  6)*  =  aa  —  2a&  -|-6&, 
we  shall  have  (a  —  1)*  =  aa  —  2a  +  1.  The  square  of  sl  —  1 
is  found,  therefore,  by  subtracting  from  aa  the  sum  of  the  two 
roots  a  and  a  —  1  namelij,  2a  —  1.  Let,  for  example,  a  =  50, 
wc  have  aa  =  2500,  and  a  —  1  =  49  :  then  49^  =  2500  —  99 
=  2401. 

272.  What  we  have  said  may  be  also  confirmed  and  illustrat- 
ed by  fractions.  For  if  we  take  as  the  i^ot  |  + 1  (which  make 
1)  the  squares  will  be  : 

9      _i       4      _l12   —  25      fljaf  IQ    1 

Further,  the  square  of  J  --  ^  (or  of  i)  will  be  |  —  4  -hi 


1 


!.  When  the  root  consists  of  a  greater  number  of  terms, 
the  method  of  determining  tlie  square  is  the  same.  Let  us  find, 
for  example,  the  square  of  a  +  b  +  c. 

a  4.  &  -f-  c 

a  +  6  -f-  c 


aa  -hab-h-ac  +  be 

ab  +  ac  +  &&  -f-  6c  -f-  cc 

aa  -f-  2a&  -|-  2ac  -f  &&  4-  26c  -f-  cc. 


n 


Chap,  ti,  Cff  Cmnponnd  Quantities,  83 

We  see  that  it  IncludeSf  first,  the  s^vure  of  each  fenn  of  the  rooty 
and  beside  that,  the  double  products  of  those  terms  multiplied  two  bij 
two, 

9.7 A.  To  illustrate  this  by  an  example,  let  us  divide  tlie  num- 
ber 256  into  three  parts,  200  -f  50  +  6  ;  its  square  will  then  be 
composed  of  the  following  parts  : 

40000  256 

2500  256 

36  

20000  1536 

2400  1280 

600  512 


655S6  655S& 

which  is  evidently  equal  to  the  product  of  256  x  256. 

275.  When  some  terms  of  the  root  are  negative,  the  square  is 
stiU  found  hj  the  same  nde  ;  but  we  must  take  care  what  signs  we 
pe^ix  to  the  double  products.  Thus,  the  square  of  a  —  b  —  c 
being  aa  +  bb  +  cc  —  2a6  —  2ac  -|-  26c,  if  we  represent  the 
number  256  by  300  —  40  —  4,  we  shall  have, 

Positive  Parts.  Negative  Parts. 


V 

+  90000 

1600 

320 

16 

+  91936 

—  26400 

—  24000 

—  2400 

—  26400 


65536,  the  square  of  256,  as  beXbre. 


84  Als^ehra,  Sect.  S. 


^^ 


CHAPTER  Yll. 

Of  lite  Exiraclion  of  Roots  applied  to  Compmind  Quantities, 

'276.  Ix  order  to  give  a  certain  rule  for  this  operation,  we 
must  consider  attentively  the  sijuare  of  the  root  a  -j-b,  wiiich  is 
aa  +  2ab  -|-  bh,  tliat  we  may  reciprocally  find  the  root  of  a 
given  square. 

Q77,  Vfe  must  consider  therefore,  first,  that  as  the  square  au 
-f  Slab  +  bb  is  composed  of  several  terms,  it  is  certain  that  the 
root  also  will  comprize  more  than  one  term ;  and  that  if  we 
write  the  square,  in  such  a  manner  that  the  powers  of  one  of  the 
letters,  as  a,  may  go  on  continually  diminishing,  the  first  term 
will  be  the  square  of  the  first  term  of  the  root.  And  since,  in 
the  present  case,  the  first  term  of  the  square  is  aa,  it  is  certain 
that  the  first  term  of  the  root  is  a, 

278.  Having  therefore  found  the  first  term  of  the  root,  that 
is  to  say  a,  we  must  consider  the  rest  of  the  square,  namely 
2ab  -f  bb,  to  see  if  we  can  derive  from  it  the  second  part  of  the 
root,  which  is  b.  Now  this  remainder  2ab  -f  bb  may  be  repre- 
sented by  the  product,  (2a  +  b)b.  Wherefore  the  remainder 
having  two  factors  2a  +  &  and  b,  it  is  evident  that  we  shall  find 
the  latter,  b,  which  is  the  second  part  of  the  root,  by  dividing 
tiic  remainder  2a6  -f  bb  by  2a  -j^  b, 

279.  So  that  the  quotient,  arising  from  the  division  of  the 
above  remainder  by  2a  -f  6,  is  the  second  term  of  the  root  re- 
quired. Now,  in  this  division  we  observe,  that  2a  is  the  double 
of  the  first  term  a,  which  is  already  determined.  So  that 
although  the  second  term  is  yet  unknown,  and  it  is  necessary, 
for  the  present,  to  leave  its  place  empty,  we  may  nevertheless 
attempt  the  division,  since  in  it  we  attend  only  to  the  first  term 
'2a,  But  as  soon  as  the  quotient  is  found,  which  is  here  6,  we 
must  put  it  in  the  empty  place,  and  thus  render  the  division 
Complete. 

280.  The  calculation,  therefore,  by  which  we  find  the  root  of 
the  square  aa  -f  Slab  -f  bb,  may  be  represented  thus  : 


Chap.  7.  Of  Compound  ^uantitie»,  85' 

aa  -^Qab  -^bb  {a  +b 
aa 


2a  +  b)  2ab  -^  bb 
2ab  -f  bb 

0. 
281.  We  may,  in  the  same  manner,  find  the  square  root  of 
other  compound  quantities,  provided  they  are  squares,  as  the 
following  examples  will  shew. 

aa  +  6ab  +  966  (a  +  36 
aa 


2a  +  36)  6fl6  -f  966 
6ab  -f  966 


0. 


4aa  —  4ab  +  66  (2a  —  6 
4aa 


4a  —  6)  —  4a6  +  66 
—  4a6  +  66 


0. 


9pp  +  24pq  -f  I6qq  {3p  +  4q 
9pp 


6p  +  4^)  24;?^  +  16qq 
9.4pq  +  I6qq 


Q5xx  —  60^  +  36  {Soc  —  6 


lOx  —  6)  —  60a;  +  36 
—  60x  -f  36 


0. 


S6  Mgehriu  Sect,  ^. 

£82.  When  there  is  a  remainder  after  the  division,  it  is  a 
proof  that  the  root  is  composed  of  more  than  two  terms.  We 
then  consider  the  two  terms  already  found  as  forming  the  first 
part,  and  endeavour  to  derive  the  other  from  jthe  remainder,  in 
the  same  manner  as  we  found  tlie  second  term  of  the  root.  The 
following-  examples  will  render  this  operation  more  clear. 

aa  -f  2ab  —  2«c  —  26c  -f  66  -f.  cc  (a  +  6  —  c 

aa 


2a  +  6) 

2fl6- 
2a6 

-  2ac  — 

.  26c  +  66  + 

cc 

+  66 

la  -f  26  - 

-c)- 

-  2ac— • 

-  2ac  — 

-  26c  -j-  cc 
'  2bc  4-  cc 

0. 


a^  +  2a^  +  Saa  -f-  2ft  +  1  (aa  +  a  +  1 


2aa  -f-  ft)  2a  =^  +  3aa 
2a3  -f-    aa 


laa  -f  2a  +  1)  2aa  -f  2a  +  1 
2aa  4-  2<i  -f  1 


0. 


a4  —  4a3ft  ^  8a63  ^  4^,4  (^aa  —  2a6  —  26* 
a* 


2aa  —  2a6)  —  4a36  +  8a63  ^  4^4 
—  4a3  6  +  4aa66 


2aa  —  4a6  —  266)  —  4aa66  +  8a63  +  46* 
—  4aa66  +  806^  4-  46* 


Chap.  7.  Of  Compmind  Quantities.  8" 

a«  — 6a«6-f  15a*56  —  20a3  63  +  I5aa6*  —  6a6«  +6^ 
a«  (a  '  —  3fl«5  -f  3a6&  —  ?»* 


—  6a*  6+    9a*6& 


2a3  ^  6aab'\-5abb)  6aHb--.Wa^b^  +  ISaab^ 


M^^6aab -{.6abb  —  b^)  —    Qa^b^+eaab"^  —  6ab' -hb^ 

—    Qa^b^  -{.6aab*  —  6ab^  -j-b^ 


0. 
283.  We  easily  deduce  from  the  rule  which  we  haVe  explain- 
ed, the  method  which  is  taught  in  books  of  arithmetic  for  tho 
extraction  of  the  square  root.     Some  examples  in  numbers ; 


529  (23 
4 

1764 
16 

(42 

2304  (48 
16 

43)  129 
129 

82)  164 
164 

88] 

1  704 
704 

0. 

0. 

0. 

•   • 
4096  ( 
36 

:64 

9604  (98 
81 

124)  496 
496 

188)  1504 
1504 

0. 

0. 

•  .  • 
15625 
1 

(125 

189) 

998001  (999 

81 

22)  56 
44 

1880 
1701 

245)  1225 
1225 

1 

989) 

17901 
17901 

0. 


88  Mgehra.  Sect.  2. 

284.  But  when  there  is  a  remainder  after  tlie  whole  operation, 
it  is  a  proof  that  the  number  proposed  is  not  a  square,  and  con- 
sequently that  its  root  cannot  be  assigned.  In  such  cases,  the 
radical  sign,  which  we  before  employed,  is  made  use  of.  It  is 
written  before  tlie  quantity,  and  the  quantity  itself  is  placed 
between  parentheses,  or  under  a  line.  Thus,  the  square  root  of 
aa  +  &6  is  represented  by  V(aa+66,)  or  by  \/aa-^bb ;  and  v(i— aro:,) 
or  vi~a;a-,  cxprcsscs  the  square  root  of  1  —  xx.  Instead  of 
this  radical  sign,  we  may  use  the  fractional  exponent  \,  and 

represent  the  square  root  of  aa  -f  lib,  for  instance  by  (aa  -f  ^6)2^, 
1 

or  by    aa+i6"l   ^. 

CHAPTER  VIII. 

Of  the  calculation  oj  Irrational  Quantities, 

285.  Whex  it  is  required  to  add  together  two  or  more  irra- 
tional quantities,  this  is  done,  according  to  the  method  before 
laid  down,  by  writing  all  the  terms  in  succession,  each  with  its 
proper  sign.  And  with  regard  to  abbreviation,  we  must  remark, 
that  instead  oJ  ^^T  +  \/a7  for  example,  we  write  2  a^T-,  and  that 
Y^I" —  \/a~  =  0,  because  these  two  terms  destroy  one  another. 
Thus,  the  quantities  3  -f  v/2~  a^^d  1 4-  V27  o,d.ded  together,  make  4  -f 
2  vsT  or  4  -f-  x/8~;  the  sum  of  5  -f  vs"  and  4  —  ^Ji  is  9  ;  and 
that  of  2  v/3"  +  3  V2~  and  V3~  —  \/2"  is  3  V3~  +  2  v/sT 

286.  Subtraction  also  is  very  easy,  since  we  have  only  to  add 
the  proposed  numbers,  changing  first  their  signs  :  the  following 
example  will  shew  this  :  let  us  subtract  the  lower  number  from 
the  upper. 

4  —    v/2~  +   2  V3~—  3  v/5"  +  4  v/s" 
1  -f  2  VS"  —  2  v/r  —  5  vr  +  6  v/6" 


3  —  3  ^2"   +4V3"+  2V5"  —  2  ^6" 

^87.  In  multiplication  we  must  recollect  that  Va"  multiplied 
hy  ^r  produces  a  ;  and  that  if  the  numbers  which  follow  the  sign 
\/  are  different,  as  a  and  b,  we  have  v/ab  for  the  jjroduct  of  \/T 
multiplied  by  v'b".  After  this  it  will  be  easy  to  perform  the 
following  examples : 


Chap.  8.  Of  Compmmd  ^lantities.  89 

1+V2"  2 —     VS" 


1  +  V/2  8  +  4  V2_ 


1+2^2    +2  =  3-f2v'2  8  —  4  =  4. 

288.  What  we  have  said  applies  also  to  imaginary  quantities ; 
we  shall  only  observe  further,  that  x/HT multiplied  by  ^Z— a  P^o- 
duces  —  a. 

If  it  were  required  to  find  the  cube  of  —  I  +  V^^  we 
should  take  the  square  of  that  number,  and  then  multiply  that 
square  by  the  same  number  :  see  the  operation  : 

—  1  +  V-3 


1  — V— 3 

—  V^^3"  — 3 

1  —  2  \/^^^  —  3  =  —  2  —  2  V^^ 
—  1    +      V=3" 


2  +  2  V-3 

—  2  v=r  +  6 

2  +  6  =  8. 
289.  In  the  division  of  surds,  we  have  only  to  express  the  pro- 
posed quQ7itities  in  the  form  of  a  fraction  ;  this  may  be  then  chang- 
ed into  another  expression  having  a  rational  denominator.  For  if 
the  denominator  be  a  +  x/T^  for  example,  and  we  multiply  both 
it  and  the  numerator  by  a  —  v^  the  new  denominator  will  be 
aa  —  b,  in  which  there  is  no  radical  sign.     Let  it  be  proposed 

to  divide  3+2  x/Y  by  1  +  v/jT;  we  shall  first  have  — — l-i-. 

1+     V2 
Multiplying  now  the  two  terms  of  the  fraction  by  1  —  v'sT  w« 
shall  have  for  the  numerator  : 


X2 


9d 

Algebra. 

3  +  2  VF 

1  —     \/2~ 

3  +  2  v/2" 

—  3V2    —4 

3— v/2'--4  =  — v/s"— 1  : 

and  for  the  denominator  : 

i+va 

1— V2 

1+V2" 

—  V2    —  2 

1_2  =  _  1 

Oiiv*  rtPU 

;t  •Pn'i/^firw*   +Viov.ofrkiio   io              '^                          •     O 

Sect  2. 


;  and  if  we  again 

multiply  the  two  terms  by  —  1,  we  shall  have  for  the  numerator 
V2^  +  1*  ^^^  foi*  the  denominator  4-  1.    Now  it  is  easy  to  shew 

that  v'2~  +  1  is  equal  to  the  proposed  fraction  -^ — ^-~  ;    for 

_  1  +     v/2 

V2~  +  1  being  multiplied  by  the  divisor  1  +  v^T  thus, 
1+V2^ 

1+V2     . 


1  +V2_ 
+  V2   +2 


we  have  1+2  V2  +2  =  3  +  2  ^2. 
Another  example  ;    8  —  5  V2'  divided  by  3  —  2  V2"  makes 

^^       Multiplying  the  two  terms  of  this  fraction  by  3  +' 


3  —  2V2 

2  \/2',  we  have  for  the  numerator, 

8  — 5  VF 
3  +  2  v/2' 


24 —  15\/2 

+  16^2" — 20 

24  +  V2~  —  20  =  4  +  VS 
and  for  the  denominator. 


Chap.  8.  Of  Cyompound  Quantities,  91 


3  —  2  V2 
S  +   2  va" 

9— 6v/2~ 
+  6^2" 


9  —  8  =  +  1. 
Consequently  the  quotient  will  be  4  +  VsT    Tlie  trutli  of  thi^ 
may  be  proved  in  the  following  manner  : 

4  +    vF 

3  —  2  ^2 


12  +  3  v/2 

— .  8  V2"  —  4 


12  —  5  ^2  —  4  =  8  —  5  ^2 
290.  In  the  same  manner,  we  may  transform  such  fractions 
into  others,  that  have  rational  denominators.     If  we  have,  for 

example,  the  fraction _,  and  multiply  its  numerator  and 

5  —  2  y/6 


denominator  by  5  +  2  ye,  we  transform  it  into  this 


5  +2  V6" 
1 


=  5  +  2  VS.     ^^  like  manner,  the  fraction =:  as- 

—  1  +  V— 3 

sumes  this  form,  —^ — -^-—^  =  -J—L And  ^  __  — 

—  4  —  2  v'6  —  V5 

114-2  4y30  _ 

becomes  =  — -i: — z. —  =  11+2  v/so. 

291.  When  the  denominator  contains  several  terms,  7ve  may  in 
the  same  manner  make  the  radical  signs  in  it  vanish  one  by  one. 

Let  the  fraction  — == = =:  be  proposed  ;  we  first  mul- 

VIO  —  V2   —  V3 

tiply  these  terms  by  v/iu  +  V^  -r  V^T  ^^^  obtain  the  fraction 
~-  .    Then  multiplvins:  its  numerator  and  denom- 

5  —  2  v/6  i  .      & 

inator  by  3  +  2  ^67  we  have  5  \/To  +  1 1  V^  +  9  ^/T  +  2  v/6a. 


92  Algebra.  Sect.  2. 

CHAPTER  IX. 

Of  Cubes,  and  of  the  Extraction  of  Cube  Roots. 

£92.  To  find  the  cube  of  a  root  a  -f  b,  we  only  multiply  its 
square  aa  +  2a6  -f  bb  again  by  a  -f  6,  tbus, 
aa  -f  2ab  -f  bb 
a  -{-b 

a^  -f-  2aab  +  abb 

aab  -f.  2a6&  +  6^ 


and  tbe  cube  will  be  =z  a^  -f  3aa&  -f-  5abb  -f  6'. 

It  co^itains  therefore  the  cubes  of  the  two  parts  of  the  roots,  and, 
beside  that,  3aab  +  Sabb,  a  quantity  equal  to  (Safe)  x  (a  +  6  ;) 
that  is,  the  triple  product  of  the  two  parts,  a  and  h,  multiplied  by 
their  sum, 

293.  So  that  whenever  a  root  is  composed  of  two  lerms,  it  is 
easy  to  find  its  cube  by  this  rule.  For  example,  the  number 
5  =  3  +  2  ;  its  cube  is  therefore  27 +  8  +  18x5  =  125. 

Let  7  +  3  =  10  be  the  root ;  the  cube  will  be  343  +  27  +  63 
X  10=  1000. 

To  find  the  cube  of  36,  let  us  suppose  the  root  36  =  30  -f  6, 
and  we  have  for  the  power  required,  27000  -f  216  +  540  x  36 
=  46656. 

294.  But  if,  on  the  other  hand,  the  cube  be  given,  namely, 
a^  -\-  Saab  +  Sabb  -\-b^,  and  it  be  required  to  find  its  root,  we 
must  premise  the  following  remarks  : 

First,  when  the  cube  is  arranged  according  to  the  powers  of 
one  letter,  we  easily  know  by  the  first  term  a^,  the  first  term  a 
of  the  root,  since  the  cube  of  it  is  a*  ;  if,  therefore,  we  subtract 
that  cube  from  the  cube  proposed,  we  obtain  the  remainder, 
Saab  +  Sabb  -f-  6^,  which  must  furnish  the  second  term  of  tbe 
root. 

295.  But  as  we  already  know  that  the  second  term  is  -f  b, 
we  have  principally  to  discover  how  it  may  be  derived  from  the 
above  remainder.  Now  that  remainder  may  be  expressed  by 
two  factors,  as  {Saa  -f-  Sab  -f  bh)  x  (&) ;  if?  therefore,  we  divide 


Chap.  9.  Of  Compound  ^antities,  95 

by  3aa  +  5ah  +  hb,  we  obtain  the  second  part  of  the  root  +  b, 
which  is  required. 

296.  But  as  this  second  term  is  supposed  to  be  unknown,  the 
divisor  also  is  unknown  ;  nevertheless  we  have  the  first  term  of 
that  divisor,  which  is  sufficient ;  for  it  is  3aa,  that  is,  thrice  the 
square  of  the  first  term  already  found  ;  and  by  means  of  this,  it 
is  not  difficult  to  find  also  the  other  part,  6,  and  then  to  complete 
the  divisor  before  we  perform  the  division.  For  this  purpose, 
it  will  be  necessary  to  join  to  3aa  thrice  the  product  of  the  two 
terms,  or  3ab,  and  bb,  or  the  square  of  the  second  term  of  the  root. 

297.  Let  us  apply  what  we  have  said  to  two  examples  of  other 
given  cubes. 

I.  a^  -I- ]2aa +  48a4-64  (a  +  4 


Saa-fl2a+l6)        12aa  +  48a -|- 64 
12aa  -f  48a  +  64 


0. 


II.  a«  —  6a«  -f  15a*  —  20a3  -|-  15^2  —  6a  +  1 

a«  {aa  —  2a  +  1 


Sa'*  —.  6a3  4-  4aa)  —  Sa'^   -f  15a*  —  20a3 
—  6as  +  12a*  —  8a3 


Sa4— 12a3  +  12aa  +  3a2  — .  6a -f  1)  3a*— I2a3  4.  I5aa— 6a-f  1 

3a*  —  12a3  +  15aa—  Ga  4.  1 

0. 
298.  The  analysis  which  we  have  given  is  the  foundation  of 
the  common  rule  for  the  extraction  of  the  cube  root  in  numbers. 
An  example  of  the  operation  in  the  number  2197  : 

2197  (10  -f  3  =  13 
1000 

300  1197 
90 
9 

399  1197 


94  Mgebra, 

Let  us  also  extract  the  cube  root  of  54965783  : 


Sect.  2. 


34965783  (300  +  20  -f  7 
2700''i000 


270000 

18000 

400 

7965783 

288400 

5768000 

30720U 

6720 

49 

2197783 

313969 

£197783 

CHAPTER  X. 

Of  the  higher  Powers  of  Compound  ((iiantities. 

299.  After  squares  and  cubes  come  higher  powers,  or 
powers  of  greater  number  of  degrees.  They  are  represented 
by  exponents  in  the  manner  which  we  before  explained  :  wc 
have  only  to  remember,  when  the  root  is  compound,  to  inclose 
it  in  a  parenthesis.  Thus  (a  +  6)*  means  that  a  -f  6  is  raised 
to  the  fifth  degree,  and  (a —  6)^  represents  the  sixth  power  of 
a  —  h.  We  shall  in  this  cliapter  explain  the  nature  of  theso 
powers. 

300.  Let  a  +  6  be  the  root,  or  the  first  power,  and  the  higher 
powers  will  be  found  by  multiplication  in  the  following  manner  : 


Chap.  10.  Of  Compound  ^antities.  95 

(a  4-  6)1  =  a  +  6 
a  +  6 


a^  4-  a6 
4-06  +  66 

{a  +  by 

=  a^  4-  2a6  4-  66 
a  4-  6 

a3  +  2aa6  4-  a66 

+    aab  4-  2a66  4-  6^ 

(a  +  6)3 

=  a'  +  3aa6  4-  3a66  4-  6^ 
a    4-  6 

4 

a*  _{-  Sa3  6  -f  3aa66  +  ab^ 

4-    aJ6  4-  3aa66  4-  2ab    4-  6* 

{a  +  6)* 

=   a*  4-  4aJ6  4-  6aa66  +  4a6'  4-  6* 
a    +  6 

o5  _|_  4a4ft  ^  6a3  66  4-  4«a63  4-  ab^ 

4-    a46  4-  4a366  4-  6aab^  +  4a6*  4-  6* 

(«  +  by 

=  a«  4-  5a*6  4-  IOa5  66  4-  lOaab^  4-  5a6*  4-  6* 
a     4-  6 

aB  -r   5a«6  4-  100^66  4-  '^Oa^b^  +  Sa^b'^  4-  ab^ 
4-   a56  +  5a466  4-  lOa'63  +  lOaab*  +  5a6«4-  6^ 

(a  4-  6)s  =  a6  +  6a«6  +  15a*66  +  20a'63  +  15aa64  4-  6a6«  +  6« 
301.  The  powers  of  the  root  a  —  b  arc  found  in  the  same 
manner,  and  we  shall  immediately  perceive  that  they  do  not 
differ  fi-om  the  preceding,  excepting  that  the  2d,  4th,  6th,  &c. 
terms  are  affected  by  the  sign  minus  ; 


96  Algebra.  Sect.  2, 

(a— &)i=a  — 6 
a  —  b 


-by 

aa 

—  6 

—  ab  +  bb 

•  b^ 

-6* 

(a- 

a 

—  Qab  +  bb 

—  b 

a^ 

—  2aab  +  abb 

—  aab  -f.  2a66  — 

(a- 

=  a3 
a 

—  Saab  4-  3a66  — 

-6» 

a^ 

—  Sa^ft  +  3aabb  ~ 

—  a^ft  +  Saabb  ~ 

-a6« 

-  3ab^  -{-  6* 

(a- 

a 

—  4a*  6  +  Gaabb  — 

—  6 

-4a63  +6* 

a« 

—  4a*6  +  6a356- 

-4aab^  +  a6* 
-6aa6»-f  4a^>''  — 

(a- 

-^da^b  +  10a^bb- 
—  6 

-10aa634.5a6*- 

a« 

—  5a«&+10a*6&- 

-lOa^b^-^-lOaab* 

—  5ab^ 

+¥ 

(a—by  =a«-— 6a«6-|-15a*6&  —  20a363^15aa54__6fl55  ^^« 
Here  we  see  that  all  the  odd  powers  of  b  have  the  sign  — , 
while  the  even  powers  retain  the  sign  +.  The  reason  of  this  is 
evident ;  for  since  —  &  is  a  term  of  the  root,  the  powers  of  that 
letter  will  ascend  in  the  following  series,  —  b,  +bbf  —  b^, 
^  b*,  —  6*,  4-  b^,  &c.  which  clearly  shews  that  the  even 
powers  must  be  affected  by  the  sign  +,  and  the  odd  ones  by  the 
contrary  sign  — • 


Chap.  10.  Of  Compound  Quantities.  97 

302.  An  important  question  occurs  in  this  place  ;  namely,  how 
we  may  find,  without  being  obliged  always  to  perform  the  same 
calculation,  all  the  powers  either  of  a  +  6,  or  a  —  b. 

We  must  remark,  in  tlie  first  place,  tliat  if  we  can  assign  all 
the  powers  of  a  +  6,  those  of  a  —  b  are  also  found,  since  we 
have  only  to  change  the  signs  of  the  even  terms,  that  is  to  say, 
of  the  second,  the  fourth,  the  sixth,  kc.  The  business  then  is 
to  establish  a  rule,  by  which  any  power  o/"  a  +  b,  however  high, 
may  be  determined  without  the  necessity  of  calculating  all  the  pre^ 
ceding  ones. 

303.  Now,  if  from  the  powers  which  we  have  already  deter- 
mined we  take  away  the  numbers  that  precede  each  term,  which 
are  called  the  coefficients^  we  observe  in  all  the  terms  a  singular 
order ;  first,  we  see  the  first  term  a  of  the  root  raised  to  the  power 
which  is  required  ;  in  tJie  following  terms  the  powers  of  a  diminish 
continually  by  unity,  and  the  powers  of  b  increase  in  the  same 
proportion  ;  so  that  the  sum  of  the  exponents  of  a  aud  of  b  is 
always  the  same,  and  always  equal  to  the  exponent  of  the  power 
required ;  and,  lastly,  we  find  the  term  b  by  itself  raised  to  the 
same  power.  If,  tlierefore,  the  tenth  power  of  a  -f  &  were 
required,  we  arc  certain  that  the  terms,  without  their  coefficients 
would  succeed  each  other  in  the  following  order;  a^^,  a^b, 
a^b^,  an^,  a^h^,  a«6S  a*6S  a^b"^,  aH\  ab%  b^\ 

304.  It  remains  therefore  to  shew,  how  we  are  to  determine 
the  coefficients  which  belong  to  those  terms,  or  tlie  numbers  by 
which  they  are  to  he  multiplied.  Now,  with  respect  to  the  first 
term,  its  coefficient  is  always  unity  ;  and  with  regard  to  the 
second,  its  coefficient  is  constantly  (lie  exponent  of  the  power  ;  hut 
with  regard  to  the  other  terms,  it  is  not  so  easy  to  observe  any 
order  in  their  coefficients.  However,  if  we  continue  those  coeffi- 
cients, we  shall  not  fail  to  discover  a  law,  by  which  we  may 
advance  as  far  as  we  please.  This  the  following  table  will 
shew. 


S«  Algebra.  Sect.  2. 

Coefficients, 

1>1 
1,  2,  1 

1,  3,  3,  1 
1,4,6,4,1 
1,  5,  10,  10,  5,  1 
1,  6,  15,  20,  15,  6,  J 
1,  7,  21,  35,  35,  21,  7,  1 
1,  8,  28,  56,  70,  5Q,  28,  8,  1 
1,  9,  56,  84,  126,  126,  84,  SQ,  9,  1 
1,  10,  45,  120,  210,  252,  210,  120,  45,  10,  1,    &c. 
We  see  then,  that  the  tenth  power  of  g  -f  6  will  be :    a*°  + 
10a»6  -f  A5a^hh-\.  120a'^b^  +  QlOa^b'^  +  252a*6*   +  210a466 
+  120a3ft7  ^  45aa68  +  lOafc^  +  b^\ 

305.  ?ri//i  regard  to  the  coefficients,  it  must  be  observed,  that  for 
each  power  their  sum  must  be  equal  to  the  number  2  raised  to  the 
same  power.  Let  a  =  1  and  6=1,  each  term,  without  the 
coefficients,  will  be  =  1 ;  consequently,  the  value  of  the  power 
will  be  simply  the  sum  of  the  coefficients  ;  this  sum,  in  the  pre- 
ceding example,  is  1024,  and  accordingly  (1  +  1)^0  =  Q^^ 
=  1024. 

It  is  the  same  with  respect  to  other  powers ;  we  have  for  the 
I.  1  +  1  =  2  =  21, 
II.  1  +  2  +  1  =  4  =  2% 

III.  1  +  3 -f- 3  +  1  =  8  =  2S 

IV.  1  +  4  +  6  +  4  +  1  =  16  =  2S 

V.   1  +  5  +  10  -f  10  +  5  +  1  =  32  =  2^, 
VI.   1  +  6  +  15  +  20  4-  15  -f  6  +  1  =  64  =  2« 
VII.   1+7  +  21  +  35+35+21+7  +  1  =  128  =2%  &c. 

306,  Anotl^er  necessary  remark,  with  regard  to  the  coeffi- 
cients, is,  that  they  increase  from  the  beginning  to  the  middle, 
and  then  decrease  in  the  same  order.  In  the  even  powers,  the 
greatest  coefficient  is  exactly  in  the  middle;  but  in  the  odd 
powers,  two  coefficients,  equal  and  greater  tlian  the  others,  are 
found  in  the  middle,  belonging  to  tlie  mean  terms. 

The  order  of  the  coefficients  deserves  particular  attention; 
for  it  is  in  tliis  order  that  we  discover  the  means  of  determining 
tliem  for  any  power  whatever,  without  calculating  all  the  pre- 


I 


Chap.  10.  Of  Compound  Quantities,  99 

ceding  powers.  We  shall  explain  this  method,  reserving  the 
demonstration  however  for  the  next  chapter. 

307.  In  order  to  find  the  coefficients  of  any  power  proposed  (the 
seventh  f  for  example,)  let  us  write  the  following  fractions  onejafter 
the  other  ; 

7        0       5       4       3       S        1 
19    ■%9    75     4>    T>    Si    y. 

In  this  arrangement  we  perceive  tliat  the  numerators  begin  by  the 
exponent  of  the  power  required,  and  that  they  diminish  successively 
by  unity  ;  while  the  denominators  foUow  in  the  natural  order  of  the 
numbers,  1,  2,  3,  4,  <SfC,  Mw,  the  first  coefficient  being  always  1, 
the  first  fraction  gives  the  second  cofficient.  The  2)roduct  of  the 
two  first  fractions,  nmltiplied  together,  represents  the  third  coefficient. 
The  product  of  the  three  first  fractions  represents  the  fourth  coeffi- 
cient, and  so  on. 

So  that  the  first  coefficient  =  1  ;  the  second  =  JJ.  =  7 ;  the 
third  =  »  X  J  =  21 ;  the  fourth  =  ^  X  |  x  f  =  35  ;  the  fifth 
=  T  ^  ¥  X  4  X  T  =  ^^  5  the  sixth  =^XjX|xJx  |  = 
21  ;  the  seventh  =  21  X  f  =  7  ;  the  eighth  =  7  X  |  =  1. 

308.  So  that  we  have,  for  the  second  power,  the  two  fractions 
|,  J  ;  whence  it  follows,  that  the  first  coefficient  =  1  ;  the  second 
=  »  =  2  ;  and  the  third  =  2x1  =  1. 

The  third  power  furnishes  the  fractions  y,  |.  -J  ;  wherefore 
the  first  coefficient  =  1 ;  the  second  =  1=3;  the  third  =  3 
X  I  =  3  ;  the  fourth  =  lx|X-|  =  l. 

\Ve  have  for  the  fourth  power,  the  fractions  ±,  |,  |,  l ;  con- 
sequently the  first  coefficient  =  1 ;  the  second  ^  =  4  ;  the  third 
*  X  I  =  6 ;  the  fourth  |  X  |  X  |  =  4  ;  and  the  fifth  4  x  ^  X  4 

xi  =  i. 

309.  This  rule  evidently  renders  it  unnecessary  for  us  to  find 
the  preceding  coefficients,  and  enahles  us  to  discover  imme- 
diately the  coefficients  which  belong  to  any  power.  Thus,  for 
the  tenth  power,  we  write  the  fi-actions  \^,  J,  |,  J,  «,  f ,  4,  |, 
|,  ^^,  by  means  of  which  we  find 

the  first  coefficient  =  1, 
the  second  =  ^  =  10, 
the  third  =  10  x  |  =  45, 
the  fourth  =  45  x  4  =  120, 
the  fifth  =  120  X  J  -  210, 


100  Mgebra.  Sects, 

the  sixth  =  210  x  |  =  252, 

the  seventh  =  252xf  =  210, 

the  eighth  =  210  x  ^  =  120, 

the  ninth  =  120  x  |  =  45, 

the  tenth  =  45  x  f  =  10, 

the  eleventh  =  10  x  -jV  =  1' 
310.  We  may  also  write  these  fractions  as  they  are,  without 
computing  their  value ;  and  in  this  way  it  is  easy  to  express 
any  power  of  a  -f  6,  however  high.  Thus,  the  hundredth  power 

offl  +  ftwillhe  (a  4-6)100  =za^^^  +  ^^o  x   a^^6+  l^^iii^ 
\      i     /  T^      1  '1x2 

■fg^afe.^  100X99X98  100x99x98x9r  , 

^  ^       1X2x3  ^        1X2X3X4  ^' 

&c.  whence  the  law  of  the  succeeding  terms  may  be  easily 
deduced. 


CHAPTER  XL 

Of  the  Transposition  of  the  Letters,  on  which  the  demonstration  of 
the  preceding  Rule  is  founded. 

311.  If  we  trace  back  the  origin  of  the  coefficients  which  we 
have  been  considering,  we  shall  find,  that  each  term  is  presented, 
as  many  times  as  it  is  possible  to  transpose  the  letters,  of  which 
that  term  consists ;  o}^  to  express  the  same  thing  differently, 
the  coefficient  of  each  terra  is  equal  to  the  number  of  transposi- 
tions that  the  letters  admit,  of  which  that  term  is  composed.  In 
the  second  power,  for  example,  the  term  ab  is  taken  twice,  that 
is  to  say,  its  coefficient  is  2 ;  and  in  fact  we  may  change  the 
order  of  the  letters  which  compose  that  term  twice,  since  we 
may  write  ab  and  ba ;  the  term  aa,  on  the  contrary,  is  found 
only  once,  because  the  order  of  the  letters  can  undergo  no 
change,  or  transposition.  In  the  third  power  of  a  +  6,  the 
term  aab  may  be  written  in  three  different  ways,  aab,  aba,  baa ; 
thus  the  coefficient  is  3.  Likewise,  in  the  fourth  power,  the 
term  a^h  or  aaab,  admits  of  four  different  arrangements,  aaabf 
aaba,  abaa,  baaa;  therefore  its  coefficient  is  4.  The  term  aabb 
admits  of  six  transpositions,  aabb,  abha,  baba,  abab,  bbaa,  baab, 
and  its  coefficient  is  6,     It  is  the  same  in  all  cases. 


J^ 


Chap.  11.  ^f  Com/pound  Quantities.  101 

312.  In  fact,  if  we  consider  that  the  fourth  power,  for  exam- 
ple, of  any  root  consisting  of  more  than  two  terms,  as  (a  +  6  + 
c  +  d)'*,  is  found  by  multiplying  the  four  factors,  I.  a  +  6  -f  c 
4.d;  II.  fl-f&  +  c  +  rf;  III.  a  +  ft  +  c  +  d;  IV.  a-f-6  +  c-|- 
d ;  we  may  easily  see,  that  each  letter  of  the  first  factor  must 
be  multiplied  by  each  letter  of  the  second,  then  by  each  letter  of 
the  third,  and,  lastly,  by  eacli  letter  of  the  fourth. 

Each  term  must  therefore  not  only  be  composed  of  four  letters, 
but  also  present  itself,  or  enter  into  the  sum,  as  many  times  as 
those  letters  can  be  differently  arranged  with  respect  to  each 
other,  whence  arises  its  coefficient. 

313.  It  is  therefore  of  great  importance  to  know,  in  how 
many  different  ways  a  given  number  of  letters  may  be  arranged. 
And,  in  this  inquiry,  we  must  particularly  consider,  whether 
the  letters  in  question  are  the  same,  or  different.  When  they 
are  the  same,  there  can  be  no  transposition  of  them,  and  for  this 
reason  the  simple  powers,  as  a^,  a^,  a*,  &c.  have  all  unity  for 
the  coefficient. 

314.  Let  us  first  suppose  all  the  letters  different ;  and  begin- 
ning with  the  simplest  case  of  two  letters,  or  ah,  we  immedi- 
ately discover  that  two  transpositions  may  take  place,  namely, 
ah  and  ha. 

If  we  have  three  letters,  ahc,  to  consider,  we  observe  that 
each  of  the  three  may  take  the  first  place,  while  the  two  others 
will  admit  of  two  transpositions.  For  if  a  is  the  first  letter,  wo 
have  two  arrangements  ahc,  ach  ;  if  h  is  in  the  first  place,  we 
have  the  arrangements  hac,  hca ;  lastly,  if  o  occupies  the  first 
place,  we  have  also  two  arrangements,  namely,  cah,  cha.  And 
consequently  the  whole  number  of  arrangements  is  3  x  2  =  6. 

If  there  are  four  letters,  ahcd,  each  may  occupy  the  first  place ; 
and  in  each  case  the  three  others  may  form  six  different  ar- 
rangements, as  we  have  just  seen.  The  whole  number  of 
transpositions  is  therefore  4  x6=  24  =  4x3x2x1. 

If  tliere  are  five  letters,  ahcdc,  each  of  the  five  must  be  the 
first,  and  the  four  others  will  admit  of  twenty  four  transposi- 
tions; so  that  the  whole  number  of  transpositions  will  be  5  x  24 
=  120  =  5X4X3X2X1. 

315.  Consequently,  however  great  the  number  of  letters  may 
be,  it  is  evident,  provided  they  are  all  different,  that  we  may 


102  Algebra.  Sect.  2. 

easily  determine  the  number  of  transpositions,  and  that  we  may 
make  use  of  the  following  table  : 

Number  of  Lettei^s.  Number  of  Transpositions. 

^^ ^ f  y -y. / 

I.  1=1. 

II.  2X1=2. 

III.  3X2X1  =  6. 

IV.  4  X  3  X  2  X  1  =  24. 
V.                                              5X4X3X2X1  =  120. 

yi.  6X5X4X3X2X1  =  720. 

Vll.  7X6X5X4X3X2X1  =  5040. 

VIII.  8X7X6X5X4X3X2X1=  40320. 

IX.  9'X  8X7X6X5X4X3X2X1  =  362880. 

X.    10X9X8X7X6X5X4X3X2X1  =  3628800. 

316.  But,  as  we  have  intimated,  the  numbers  in  this  table 
can  be  made  use  of  only  when  all  the  letters  are  different ;  for 
if  two  or  more  of  them  are  alike,  the  number  of  transpositions 
becomes  much  less  ;  and  if  all  the  letters  are  the  same,  we  have 
only  one  arrangement.  We  shall  now  see  how  the  numbers  in 
the  table  are  to  be  diminished,  according  to  the  number  of  letters 
that  are  alike. 

317.  When  two  letters  arc  given,  and  those  letters  are  the 
same,  the  two  arrangements  are  reduced  to  one,  and  conse- 
quently the  number,  which  we  have  found  above,  is  reduced  to 
the  half;  that  is  to  say,  it  must  be  divided  by  2.  If  we  have 
three  letters  alike,  the  six  transpositions  are  reduced  to  one ; 
whence  it  follows  that  the  numbers  in  the  table  must  be  divided 
by  6  =  3x2x1.  And  for  the  same  reason,  if  four  letters  are 
alike,  we  must  divide  the  numbers  found  by  24  or  4  x  3  x  2 
X  1,  &c. 

It  is  easy  therefore  to  determine  how  many  transpositions  the 

letters  aaabbc^  for  example,  may  undergo.    They  are  in  number 

6,  and  consequently,  if  they  were   all   different,  they  would 

admit  of  6x5x4x3x2x1  transpositions.     But  since  a  is 

found  thrice  in  those  letters,  we  must  divide  that  number  of 

transpositions  by  3  x  2  x  1  ;  and  since  b  occurs  twice,  we  must 

again  divide  it  by  2  x  1  ?  the  number  of  transpositions  required 

Ml   XI-       p         u  6  X  5  X  4  X  ">  X  '^  X  1        ^  ,    .   ,.  ^        an 

Will  therefore  be  =  -«- — - — - — - — — -  =  5x4x3  =  60. 
3X^X1X2X1 


Chap.  11.  Of  Compmind  ^lantities,  103 

318.  It  will  now  be  easy  for  us  to  determine  the  coefficients 
of  all  the  terms  of  any  power.  We  shall  give  an  example  of  the 
seventh  power  (a  +  6)'. 

The  first  term  is  a^,  which  occurs  only  once  ;  and  as  all  the 

other  terms  have  each  seven  letters,  it  follows  that  the  number 

of  transpositions  for  each  term  would  be  7x6x5x4x3x2 

X  1>  if  all  the  letters  were  different.     But  since  in  the  second 

term,  a^b,  we  find  six  letters  alike,  we  must  divide  the  above 

product  by  6x5x4x3x2x1,  whence  it  follows  that  the 

^  .      ,.  7x6x5x4x3x2x1        - 

coemcieBt  js  =  — :: ; — ;; ; —  =  \. 

.,  6X5X4x3x2x1  ' 

*^1rn  the  third  term  a^bb,  we  find  the  same  letter  a  five  times, 

and  the  same  letter  b  twice;  we  must  therefore  divide  that 

number  first  by  5x4x3x2x1,  and  then  also  by  2  x  1 ; 

,        ,.   .,     «,  ..7x6x5x4x3x2x1  7x6 

whence  results  the  coethcient  - — — ; ■  = -. 

5X4X3X2X1X^X1       1x2 

The  fourth  term  a^h^  contains  the  letter  a  four  times,  and  the 
letter  b  thrice  ;  consequently,  the  whole  number  of  the  transpo- 
sitions of  the  seven  letters,  must  be  divided,  in  the  first  place, 
by  4  X  3  X  2  X  1^  and,  secondly,  by  3  x  2  x  1,  and  the  second 
7  X  5  X  5  X  4  X  3  X  ^'  X  I  7  X  f^  X  !> 


coefficient  becomes  = 


4X3X2X1X3X2X1     IX^iXo' 


In  the  same  manner,  we  find  — — - — ^  for  the  coefficient 

1X2X3X4 

of  the  fifth  term,  and  so  of  the  rest ;  by  which  the  mle  before 
given  is  demonstrated. 

319.  These  considerations  carry  us  further,  and  shew  us  also, 
how  to  find  all  the  powers  of  roots  composed  of  more  than  two 
terms.  We  shall  apply  tliem  to  the  third  power  of  a  -f  6  +  c  ; 
the  terms  of  which  must  be  formed  by  all  the  possible  combina- 
tions of  three  letters,  each  term  having  for  its  coefficient  the 
number  of  its  transpositions,  as  above. 

"\>*ithout  performing  the  multiplication,  the  third  power  of 
(a  -f.  6  4.  c)  will  be  a»  +  Saab  +  Saac  -|-  3abb  -f-  6abc  -f  5acc  + 
63  4.  3hb  -f-  3&cc-f-  c». 

Suppose  a  =  1,  6  =  l,  c  =  1,  the  cube  of  1  +  1  4- 1,  or  of  3, 
will  be  l  +  a-fS-|-3-f.6  +  3-fl  +  SH-3  +  l=:27. 

This  result  Ls  accurate,  and  confirms  the  rule. 


IW  Mgehra,  Sect.  2. 

If  we  had  supposed  a  =  1,  6  =  1,  and  c  =  —  l,  we  should 
have  found  for  the  cube  of  1  +1  —  I,  that  is  of  1, 

1  +  3— 3  +  3  —  6  +  3  +  1— .3  +  3— 1  =  1. 


CHAPTER  XII. 

Of  the  expression  of  Irrational  Powers  hy  Infinite  Series, 

320.  As  we  have  shewn  the  method  of  finding  any  power  of 
the  root  a  +  6,  however  great  the  exponent,  we  are  able  to 
expr-ess,  generally,  the  power  of  a  +  5,  whose  exponent  is  unde- 
termined. It  is  evident  that  if  w^e  represent  that  exponent  hy 
71,  wc  shall  have  by  the  rule  already  given  (art.  307  and  the 
following)  : 

n — 2    „  o, ,    .    n       n — 1       n — 2       n — ^3        ,, , 

__  a"-3ft3  +-X    -^  X    -g-X    -^  a^^-^h^  +  &c. 

321.  If  the  same  power  of  the  root  a  —  h  were  required,  we 
should  only  change  the  signs  of  the  second,  fourth,  sixth,  &c. 

terms,  and  should  have  (a  —  bY=  a^  —  —  a"-^6  +  —  X  ^^ 
^  >  1  ^   I  ^     2 

t,  91.9         '«      ??— "1      ^ — 2    ^  ,j ,    .    n       n — 1       n — 9         n — 3 
fl"-2&8 7  X  — r  X  -^  a«^Z>3  ^     ^  ^  X— r 

12  3  12  3  4 

322.  These  formulas  are  remarkably  useful ;  for  they  serve 
also  to  express  all  kinds  of  radicals.  AYe  have  shewn  that  all 
irrational  quantities  may  assume  the  form  of  powers  whose 

2  1  3  _  \ 

exponents  are  fractional,  and  that  \/a  =  a^  ;  ^a   =  a^,   and 

4    1 

y/a  =  a*j  &c.     We  have,  therefore,  also, 

V  (a  +  6)  =  (a  +6)S-  V  C«  +  ^)  =  («  +  # 

and  y/  (fl  +  fc)  =  (a  +  6)*,  &c. 
Wherefore,  if  we  wish  to  find  the  square  root  of  o  +  &,  we 
have  only  to  substitute  for  the  exponent  n  the  fraction  |,  in  the 
general  formula,  [art.  320,]  and  we  shall  have  first,  for  the- 
coefficients. 


5 

n — 3 
4 

=  — 

5, 

8' 

n — 4 

'      5     ■ 

= 

ai  = 

=  v/J" 

and 

a""^ 

*      1 

> 

Chap.  12.  Of  Compound  Quantities,  105 

«  ___     1  ^  n — 1  _         1  ^  n — 2  __ 
7""    I'T   ~""~4'    "3~" 

;  = .     Then,  a' 

10'      6  12 

a"~^  =  — =r- :  a"~^  = =,  &-c.  or  we  miffht  express  those 

powers  of  a  in  the  following  manner ;  a"  =  ^/Z  ;  a""^  =  — —  ; 

,  323.  This  heing  laid  down,  the  square  root  of  a  +  6,  may  be 
expressed  in  the  following  manner  : 

324.  If  a  therefore  be  a  square  number,  we  may  assign  the 
value  of  y/7l  and,  consequently,  the  square  root  of  a  -f  6  may  be 
expressed  by  an  infinite  scries,  without  any  radical  sign. 

Let,  for  example,  a  =  cc,  we  shall  have  -^/^  =  c;  then  v^  (cc 

,.  1         /;  Ibb        I       h^  5        h^    . 

+  b)  =c+-  X  -  -  ^^-,  +  p,X-  -  —  X-,  kc. 

TVe  see,  therefore,  that  there  is  no  number,  whose  square 
root  we  may  not  extract  in  the  same  waj^ ;  since  every  number 
may  be  resolved  into  two  parts,  one  of  which  is  a  squai'c  repre- 
sented by  cc.  If  we  require,  for  example,  tiie  square  root  of  6, 
we  make  6=4+2,  consequently  cc  =  4,  c  =  2,  ft  =  ,2,  whence 
results  vr  =  2  +  ^  —  ^V  +  A  —  t A 4»  ^c. 

If  we  take  only  the  two  leading  terms  of  this  series,  we  shall 
1 
have  2^  =  *,  the  square  of  which  '/  is  ^  greater  than  6 ;  but 

if  we  consider  three  terms,  we  have  24§-  =  4t>  ^^'®  square  of 

which,  y^V,  is  still  ,y^  too  small. 

325.  Since,  in  this  example,  |  approaches  very  nearly  to  the 
true  value  of  veT  we  shall  take  for  6  the  equivalent  quantity  y 
—  1.     Thus  cc=:y  ;  c  =  | ;  6  =  J  ;  and  calculating  only  the 

two  leading  terms,  we  find  V6~  =  f  +i  X  ^^^^  =  |  —  |  x  —  = 

14 


106  Algebra.  Sect.  2. 

•J  —  ^V  =  IJ  J  *^6  square  of  this  fraction,  being  WV>  exceeds 
the  square  of  v^e"  only  by  ^ J^. 

Now,  making  6  =  \%\^  —  ^J^,  so  that  c  =  | J  and  6  =  — 
•jj^ ;  and  still  taking  only  the  two  leading  terms,  we  have  ^6" 

—  4  9    J_    1    V   —    *^^    4  9  1    V   ^i^O^    4  9   1  4  8  0  1 

—  If^  -r  5  A  -7^  —  SIT  S  ^  T9~  ^   'SH  1T6-U    —   TTTTT' 

Ti'5  20" 

the  square  of  which  is  «^yj*j.y^y.     Now  6,  when  reduced  to 
the  same  denominator,  is  =  y^-^gV/  »  the  error  therefore  is 

only  fT4TT15-Tr* 

326.  In  the  same  manner,  we  may  express  the  cube  root  of 

3  1 

a  +  h  by  an  infinite  series.    For  since  v/  («  +  ^)  =  («  +  V)'^9 
we  shall  have  in  the  general  formula  w  =  ^,  and  for  the  coeffi- 

cients,J^=  i;'iZl}=-i;!!=.^  =  _i;   'tr_5==Ji,. 
'   1  3  '      2  3'      3  9'       4  3' 

— —  =  —  — ,  &c.  and  with  regard  to  the  powers  of  a,  we  shall 

3    _  3  _  3  _ 

have  a«  =  v^;  a"-^  =  :5^  ;  a"-^  =  :^  ,•  a""^  =  :^,  &c.  then 

a  aa  a^ 

3  a  9  aa         81  a^ 

J^xb*y^,  &c. 

^43  a^ 

s_ 

327.  If  a  therefore  be  a  cube,  or  a  =  c^,  we  have  \/a  =  c,  and 

the  radical  signs  will  vanish ;  for  we  shall  have 

3    ,  ,    ,    ,.  1  b  I        bb        5        b^  10 

328.  We  have  therefore  arrived  at  a  formula,  which  will 
enable  us  to  find  by  approximation,  as  it  is  called,  the  cube  root 
of  any  number  ;  since  every  number  may  be  resolved  into  two 
parts,  as  c^  -f  5,  the  first  of  which  is  a  cube. 

If  we  wish,  for  example,  to  determine  the  cube  root  of  2,  we 
represent  2  by  1  +  1,  so  that  c  =  1   and  &  =  1,  consequently 

^2"  =  I  +  -I  —  1^  +  TT'  ^^'  ti^^  t^^^  leading  terms  of  this 


Chap.  15.  Of  Compound,  ^lantities.  107 

series  make  1^  =  -J  the  cube  of  which  |^  is  too  great  by  |4* 
Let  us  then  make  2  =  |^  —  J^,  wo  have  c  =  J  and  6  =  —  ^  ^, 

and  consequently  ^2   =  4  +  i  ><  -fr*    These  two  terms  give 

4  —  /s  =  ^i,  the  cube  of  which  is  |f||U-  Now,  2  =  |^||||, 
so  that  the  error  is  ^|.|7  5  ^^  jj,  this  way  we  might  still  approx- 
imate, and  the  faster  in  proportion  as  we  take  a  greater  number 
of  terms. 


CHAPTER  XIII. 

Of  the  resolution  of  JVlegative  Powers. 

329.  We  have  already  shewn,  that  we  may  express  —  by  a~^ ; 

we  may  therefore  also  express  — -  by  {a  -f  6)-^ ;  so  that  the 

fraction   — —  may  be  considered  as  a  power  of  a  -f  6,  namely 

that  power  whose  exponent  is  —  1  ;  and  from  this  it  follows, 
that  the  series  already  found  as  the  value  of  (a  -f  by  extends 
also  to  this  case. 

330.  Since,  therefore,  — —  ij^  the  same  as  (a  +  6)-i,  let  us 

suppose,  in  the  general  formula,  n  =  —  1 ;  and  we  shall  first 

have  for  the  coefficients  —  =  —  1;  ^^  = — 1  ;  ^    ^  = i- 

1  ^  3  ' 

-—  =  —  1,  &c.    Then,  for  the  powers  of  a;  a"  z=  a~^  =  ~; 

a«-i  =  a-=  =  4  '  «""'  =  ^ '  «"-'  =  A*  &c.    So  that  (a  +  h)-^ 

_     1           1           b          bb          h^          h^          b'     .  ,  X,  .     • 

—  -77.  =  -- ^    H -, -J.   +   -T -.9  occ.  and  this  is 

the  same  series  that  we  found  before  by  division. 

331.  Further,    ^^——  being  the  same  with  (a  +  &)-2,  let  us 

reduce  this  quantity  also  to  an  infinite  series.  For  this  purpose, 
we  must  suppose  n  =  —  2,  and  we  shall  first  have  for  the  coeffi- 


% 


\9^  'Algebra.  Sect.  2. 

cients  i^  =  -^;!!Zl^=-  i.!!Z?  =  _«^.   !^3    _   _ 

1  1         ^  2^      3  S'        4        — 

--,  &c.      Then,  for  the  powers  of  a ;  a"  =  -i  ;  a"-^  =  —  • 
4  ft^  a^  ' 


a' 


;  a"~^  =  — ,  6cc.     "^e  |,herefore  obtain  (a  +  fe)-^  =:= 


^   ^    ^^    +  1    ^   -i   ^    -i   ^    1  ^    a-^'  ^^-     ^^^^''  T  =  2,. 
-|X|  =  3;  fX|x4=4;  4x|x4xf  =  5,  6cc.     Conse- 

1  1  b  b^  b^  b^ 

qiiently,  we  have   ,    ,  ,,,     =    -  —  2-  +  3-  —  4-    +  5-- 

332.  Let  us  proceed  and  suppose  n  =  —  3,  and  we  shall  have 

a  series  expressing  the  value  of  - — j—-,  or  of  (a  +  fe)"~^.   The 

^  .     .        -ii  .      3  3     n — 1  4     n — 2  5 

coefficients  will  be  —  = : = ;  = : 

1  l'      2  2        S  3 

.1^  = ,  &c.  and  the  powers  of  a  become,  a"  =  —  a""^  = 

_:  a"~^  =  — ,  &c.  which  cives  - — -p--  =  — — .  +  — 

4^2  3  4  5&3  3  4^0^6fc4  , 

X-^rs  -  -1   ^    ^    ^   1  rl^    +  1    ^    ^   ^    -^    ^     4  ^-""^ 

4.454--,  &c. 

Let  us  now  make  11=  —  4  ;  we  shall  have  for  the  coefficients 

n  _         4  ,   n — 1  _  5  ^  « — 3  _  6  ^    n — 3  _         ^    At 

T  ~"        1 '  ~i  ^ '   "i"  "■       ^ '  ~T  —       T 

1  11  1 

and  for  the  powers,  a"  =  —  :  a"~^  =  —  ;  a"-^  —      .  ^n-s  _       . 

1  i  1  4  & 

a"-^  =  -r^  &c,  whence  we  obtain  ;     ,    ,  ,,  .■   =    — 7   X   —. 

46fc2  456         63  4067ft'* 


Chap.  13.  Of  Coinpmmd  C(uantities»  109 

±  b  h^  b^  b*  b^ 

and  c=    1  —  4-  —  101-  —  20-^    +  35%  —  56^,  +,   &c. 

333.  The  different  cases  that  have  been  considereo  enable  us 
to  conclude,  with  certainty,  that  we  shall  have,  generally,  for 
any  negative  power  of  a  +  6  ; 

1             1           m          b      ,  m  ^      m+±  ^      b^           m        m+± 
■=- 7  X  -zri-.  +  -7  X  — T—  X  -;:n:. r  X   —~— 


3  a^-^^ 

And,  by  means  of  this  formula,  we  may  transform  all  such 
fractions  into  infinite  series,  substituting  fractions  also,  or 
fractional  exponents,  for  m,  in  order  to  express  irrational  quan- 
tities. 

334.  The  following  considerations  will  illustrate  this  subject 
further. 

We  have  seen  that, 

If,  therefore,  we  multiply  this  series  by  a  +  &,  the  product 
ouglit  to  be  =  1 ;  and  this  is  found  to  be  true,  as  we  shall  see 
by  performing  the  multiplication  : 

1  b        b^  b'        b^  b^    ,     SL^ 

a  +  & 


h 

b* 

/»' 

6* 

bs 

1 

a 

b 

+ 

a3 

b^ 

b^ 

b* 

b^ 

+,  &c. 

+ 

a 

"" 

a* 

+ 

a} 

a* 

+ 

as  " 

— ,  &c. 

1 

S35.  We  have  also  found,  that  -, rrr-  = -»   H — 

-I — ,  &c.  If,  therefore,  we  multiply  this  series 

a^         a^  a^  * 

by  (a  +  6)*j  the  product  ought  also  to  be  =  1.     Now  (a  -f  &)* 

=  rta  4-  2a&  +  bb.    See  the  operation  : 


11«  Mgehra.  Sect.  2. 


1    26    366     463 
aa         a'     a*     a* 

f   .  56*     66* 

aa   +  2a6  +  66 

26    366     463 
a     aa              a^ 

56*     66*  .  . 

26    466    66» 
"^    a            aa            a^ 

bb           263 

r^6*   46*    , 

1  =    the  product,  wliich  the  nature  of  the  thing  required. 
336.  If  we  multiply  the  series  which  we  found  for  the  value 

q{ __,  by  a  +  6  only,  the  product  ought  to  answer  to  the 

KP'+by 

fraction    — r?  or  be  equal  to  the  series  already  found,  namely, 

H — 1 H — -,  &c.  and  this  the  actual  multiplica- 

a         a^        a^          a*         <i«  * 

tion  will  confirm. 

1  26  366  46'  56*     „ 


a^ 


a  +  h 


1 

9h 

366 

463 

56* 

&c. 

—. 

+ 

— 

— ■ 

+ 

a 

aa 
6 

a' 

266 

a* 
36' 

a* 
46* 

+ 

aa 

a^ 

+ 

a* 

a* 

,  6cc. 

a         ac.         a3  a*    +  «>        ^ 


SECTION  THIRD. 

«F    RATIOS  AND    PROPORTIONS. 

CHAPTER.  I. 

Qf  ^Arithmetical  Ratio,  or  of  the  difference  between  two  JSPumbers. 

ARTICLE  3S7. 

Two  quantities  are  either  equal  to  one  another,  or  they  are 
not.  In  the  latter  case,  where  one  is  greater  than  the  other, 
we  may  consider  their  inequality  in  two  different  points  of  view  : 
we  may  ask,  hoiv  jtmch  one  of  the  quantities  is  greater  than  the 
other  ?  Or,  we  may  ask,  how  many  times  the  one  is  greater 
than  the  other  ?  The  results,  which  constitute  the  answers  to 
these  two  questions,  are  both  called  relations,  or  ratios.  We 
usually  call  the  former  arithmetical  ratio,  and  the  latter  geomet- 
rical ratio,  without  however  these  denominations  having  any 
connexion  with  the  thing  itself :  they  have  been  adopted  arbi- 
trarily. 

338.  It  is  evident,  that  the  quantities  of  which  we  speak  must 
be  of  one  and  the  same  kind  ;  otherwise,  we  could  not  determine 
any  thing  with  regard  to  their  equality,  or  inequality.  It  would 
be  absurd,  for  example,  to  ask  if  two  pounds  and  three  ells  are 
equal'  quantities.  So  that  in  what  follows,  quantities  of  the 
same  kind  only  are  to  be  considered  ;  and  as  they  may  always 
be  expressed  by  numbers,  it  is  of  numbers  only,  as  was  men- 
tioned at  the  beginning,  that  we  shall  treat. 

339.  When  of  two  given  numbers,  therefore,  it  is  required  to 
find,  liow  much  one  is  greater  than  tlie  other,  the  answer  to  this 
question  determines  the  arithmetical  ratio  of  the  two  numbers. 
Now,  since  tliis  answer  consists  in  giving  the  difference  of  the 


112  Mgeh-a.  Sect.  3. 

two  numbers,  it  follows,  that  an  arithmetical  ratio  is  nothing 
but  the  difference  between  two  numbers  :  and  as  tliis  appears  to 
be  a  better  expression,  we  sliall  reserve  the  words  ratio  and 
reMioiif  to  express  geometrical  ratios. 

340.  The  difference  between  two  numbers  is  found,  we  know, 
by  subtracting  the  less  from  the  greater  ;  nothing  therefore  can 
be  easier  than  resolving  the  question,  how  much  one  is  greater 
than  the  other.  So  tliat  when  the  numbers  are  equal,  the  dif- 
ference being  nothing,  if  it  be  inquired  how  much  one  of  the 
numbers  is  greater  than  the  other,  we  answer,  by  nothing.  For 
example,  6  being  =  2x3,  the  difference  between  6  and  2  x  3  is  0. 

341.  But  when  the  two  numbers  are  not  equal,  as  5  and  3, 
and  it  i .  inquired  how  much  5  is  greater  than  3,  the  answer  is, 
2  ;  and  it  is  obtained  by  subtracting  S  from  5.  Likewise  15  is 
greater  than  5  by  10  ;  and  20  exceeds  8  by  12. 

342.  We  have  three  things,  therefore,  to  consider  on  this 
subject ;  1st,  the  greater  of  the  two  numbers  ;  2d,  the  less  ;  and 
Sd,  the  difference.  And  these  three  quantities  are  connected 
together  in  such  a  manner,  that  two  of  the  three  being  given, 
we  may  always  determine  the  third. 

Let  the  greater  number  =  a,  the  less  =  6,  and  the  difference 
=  d ;  the  difference  d  will  be  found  by  subtracting  h  from  a,  so 
that  d  =  a  —  b  ;  whence  we  see  how  to  find  rf,  when  a  and  b  are 
given. 

343.  But  if  the  difference  and  the  less  of  the  two  numbers,  or 
bf  are  given,  we  can  determine  the  greater  number  by  adding 
together  the  difference  and  the  less  number,  which  gives  a  =  6 
+  d.  For,  if  we  take  from  b  -\-  d  the  less  number  b,  there 
remains  d,  which  is  the  known  difference.  Let  the  less  number 
=  12,  and  the  difference  =  8  ;  then  the  greater  number  will  be 
=  20. 

344.  Lastly,  if  beside  the  difference  d,  the  greater  number  a 
is  given,  the  other  number  b  is  found  by  subtracting  the  differ- 
ence from  the  greater  number,  whicli  gives  &  =  a  —  d.  For,  if 
I  take  the  number  a  —  d  from  tlie  greater  number  a,  there 
remains  d,  which  is  the  given  difference. 

345.  The  connexion,  therefore,  among  the  numbers  a,  ft,  tZ,  is 
of  such  a  nature,  as  to  give  the  three  following  results  :  P**  d  =  a 


Chap.  2.  Of  Compmnd  ^tantities.  113 

h;  2^  d  =  b  -^d;    2^-  b  =  a  —  d;  and  if  one  of  these  three 

comparisons  be  just,  the  others  must  necessarily  he  so  also. 
Wherefore,  generally,  [{  %  =  x  +  y,  it  necessarily  follows,  that 
y  =  %  —  X,  and  x  =  %  —  y. 

346.  \Vith  regard  to  these  arithmetical  ratios  we  must  remark, 
that  if  we  add  to  the  two  numbers  a  and  b,  a  number  c,  assumed 
at  pleasure^  or  subtract  it  from  them,  the  difference  remains  the 
same.  That  is  to  say,  if  d  is  the  difference  between  a  and  b, 
that  number  d  will  also  he  the  difference  between  a  -f.  c  and 
b  +  c,  and  between  a  —  c  and  b  —  c.  For  example,  the  differ- 
ence between  the  numhers  20  and  12  being  8,  that  difference 
will  remain  the  same,  whatever  numher  we  add  to  the  numhers 
20  and  12,  and  whatever  numbers  we  subtract  from  them. 

347.  The  proof  is  evident ;  for  if  a  —  b  =  d  we  have  also 
(a  +  c)  —  (b  -{-c)  =:d;  and  also  (a  —  c)  —  (b  —  c)  =  d. 

348.  If  we  double  the  two  numbers  a  and  h,  the  difference  will 
also  become  double.  Thus,  when  a  —  6  =  rf,  we  shall  have  2a  — 
fift  z=  2d;  and,  generally,  na  —  nb  =  nd,  whatever  value  we 
give  to  n, 

CHAPTER  IL 

Of  Arithmetical  Proportion. 

349.  When  two  arithmetical  ratios,  or  relations,  are  equal, 
this  equality  is  called  an  arithmetical  proportion. 

Thus,  when  a  —  b=zd  and  p  —  q  =  d,  so  that  the  difference 
is  the  same  between  the  numbers  p  and  q,  as  between  the  num- 
bers a  and  b,  we  say  that  these  four  numbers  form  an  arithmeti- 
cal proportion  ;  which  we  write  thus,  a  —  b  =  p  —  g,  expressing 
clearly  by  this,  that  the  difference  between  a  and  b  is  equal  to 
the  difference  between  p  and  q. 

350.  An  arithmetical  proportion  consists  therefore  of  four 
terms,  which  must  be  such,  that  if  we  subtract  the  second  from 
the  first,  the  remainder  is  the  same  as  when  we  subtract  the 
fourth  from  the  third.  Thus,  the  four  numhers  12,  7,  9,  4,  form 
an  arithmetical  proportion,  because  12  —  7=9  —  4.  (*) 

(•)  To  «hew  tliat  these  terms  make  such  a  proportion,  some  write  them 
thus  ;    12  .  .  7  :  :  9  .  .  4. 

15 


W4  Mgehra,  Sect.  S. 

351.  When  we  have  an  arithmetical  proportion f  as  it  —  b  =  p 
*—  q,  we  may  make  the  second  and  third  change  places,  writing 
a  —  p  =  b  —  q ;  and  this  equalitij  will  he  no  less  true  ;  for,  since 
fit  —  b  z=zp  —  q,  add  h  to  both  sides,  and  we  have  a  =  h  +p  —  q; 
then  subtract  j?  from  both  sides,  and  we  have  a  — p  z=zh  —  q. 

In  the  same  manner,  as  12  —  7  =  9  —  4,  so  also  12  —  9  = 
r  — 4. 

352.  TFc  may,  in  etery  anthmetical  proportion,  put  the  second 
term  also  in  the  place  of  the  first,  if  we  maJte  the  same  transposi- 
tion  of  the  third  and  fourth.  That  is  to  say,  if  o  —  b  =p  —  q^ 
we  have  also  b  —  a  =  g  —  p.  For  b  —  a  is  the  negative  of 
a  —  b,  and  q  —  j?  is  also  the  negative  of  p  —  q.  Thus,  since 
12  —  7  =  9  —  4,  we  have  also,  7  —  12  =  4  —  9. 

353.  But  the  great  property  of  every  arithmetical  proportion  is 
this  ;  that  the  sum  of  the  second  and  third  term  is  always  equal  to 
the  sum  of  the  first  and  fourth.  This  property,  which  we  must 
particularly  consider,  is  expressed  also  by  saying  that,  the  sum 
of  the  means  is  equal  to  the  sum  of  the  extremes.  Thus,  since 
12  —  7=9  —  4,  we  have  7  +  9  =  12  +  4  ;  and  the  sum  we 
find  is  16  in  both. 

354.  In  order  to  demonstrate  this  principal  property,  let  a  — 
b  =  p  —  q'y  if  we  add  to  both  b  -{- q,  we  have  a  ■}-  q  =  b  -j-  p  ^ 
that  is,  the  sum  of  the  first  and  fourth  terms  is  equal  to  the  sum 
of  the  second  and  third.  And  conversely,  if  four  numbers,  a,  b,  p, 
q,  are  such,  that  the  sum  of  the  second  and  third  is  equal  to  the  sum 
of  the  first  and  fourth,  that  is,  if  6  -f-  j?  =  a  +  q,  we  conclude, 
without  a  possibility  of  mistake,  that  these  numbers  are  in  arith- 
metical proportion,  and  that  a  —  b  =  p  —  q.  For,  since  a  -{-  q 
=  &  +  ;?,  if  we  subtract  from  both  sides  &  +  g,  we  obtain  a  —  b 
=  p—q. 

Thus,  the  numbers  18,  13,  15,  10,  being  such,  that  the  sum 
of  the  means  (13  -f-  15  =  28,)  is  equal  to  the  sum  of  the  ex- 
tremes (18  +  10  =  28,)  it  is  certain,  that  they  also  form  an 
arithmetical  proportion;  and,  consequently,  that  18  —  13  = 
15  —  10. 

S55,  It  is  easy,  by  means  of  this  property,  to  resolve  the  fol- 
lowing question.  The  three  first  terms  of  an  arithmetical  pro- 
portion being  given  to  find  the  fourth  ?    Let  a,  b,  p,  be  the  three 


\ 


Chap.  2.  Of  Ctyinpound  Quantities,  115 

first  terms,  and  let  us  express  the  fourth  by  </,  which  it  is 
required  to  determine,  then  a  +  g  =  6  -f  jJ ;  by  subtracting  a 
from  both  sides,  we  obtain  q  =z  b  -\- p  —  a. 

Thus,  the  fourth  term  is  found  by  adding  together  the  second  and 
third,  and  subtracting  the  first  from  that  sum.  Suppose,  for  ex- 
ample, that  19,  28,  13,  are  the  three  first  terms  given,  the  sum 
of  the  second  and  tliird  is  =  41 ;  take  from  it  tlie  first,  which  is 
19,  there  remains  22  for  the  fourth  term  sought,  and  the  arith- 
metical proportion  will  be  represented  by  19  —  28  =  13  —  22, 
or,  by  28  —  19  =  22  —  13,  or,  lastly,  by  28  •—  22  =  19  —  13. 

356,  When  in  an  arithmetical  proportion,  the  second  term  is  equal 
to  the  third,  we  have  only  three  numbers  ;  the  property  of  w  hich 
is  this,  that  the  first,  minus  the  second,  is  equal  to  the  second, 
minus  the  third ;  or,  that  the  difference  betsveen  the  first  and 
the  second  number  is  equal  to  the  difference  between  the  second 
and  the  third.  The  three  numbers  19,  15,  11,  are  of  this  kind, 
since  19  —  15  =  15  —  11. 

S57,  Three  such  numbers  are  said  to  form  a  continued  anlh- 
metical  proportion,  which  is  sometimes  written  thus,  19  :  15  :  11. 
Such  proportions  are  also  called  arithmetical  progressions,  par- 
ticularly if  a  greater  number  of  terms  follow  each  other  according 
to  the  same  law. 

An  arithmetical  progression  may  be  either  increasing,  or 
decreasing.  The  former  distinction  is  applied  when  the  terms 
go  on  increasing,  that  is  to  say,  when  the  second  exceeds  the 
first,  and  the  third  exceeds  the  second  by  the  same  quantity ; 
as  in  the  numbers  4,  7,  10.  The  decreasing  progression  is  that, 
in  wldch  the  terms  go  on  always  diminishing  by  the  same  quan- 
tity, such  as  the  numbers  9,  5, 1. 

258.  Let  us  suppose  the  numbers  a,  b,  c,  to  be  in  arithmetical 
progression;  then  a  —  b  =zb  —  c,  whence  it  follows,  from  the 
equality  between  the  sum  of  the  extremes  and  that  of  the  means, 
that  2&  =  a  +  c ;  and  if  we  subtract  a  from  both,  we  have  c  = 
2&  —  a, 

359.  So  that  when  the  two  first  terms  a,  b,  of  an  arithmetical 
progression  arc  given,  the  third  is  found  by  taking  the  first  from 
twice  the  second.  Let  1  and  3  be  the  two  first  terms  of  an  arith- 
metical progression,  the  third  will  be  =  2x3  —  1  =  5.  And 
tltesc.tlu'ce  numbers  1,  3,  .5  give  the  proportion  1—3  =  3  —  5, 


116'  Mgeh-a.  Sect.  2. 

560.  Py  following  the  same  method,  we  may  pursue  the 
arithmetical  progression  as  far  as  we  please ;  we  have  only  to 
find  the  fourth  by  means  of  the  second  and  third,  in  the  same  man- 
ner as  we  determined  the  third  by  means  of  the  first  and  second, 
and  so  on.  Let  a  be  the  first  term,  and  h  the  second,  the  third 
will  be  =  2b  —  a,  the  fourth  =  46  —  2a  —  6  =  36  —  2a,  the 
fifth  66  —  4a  —  26  +  a  =  46  —  Sa,  the  sixth  =  86  —  6a  — •  3& 
-f  2a  =  56  —  4a,  the  seventh  =  106  —  8a  —  46  +  Sa  =  66 
-^  Soj  &c. 


CHAPTER  III. 

Of  Arithmetical  Progressions. 

561,  We  have  remarked  already,  that  a  series  of  numbers 
composed  of  any  number  of  terms,  which  always  increase,  or 
decrease,  by  the  same  quantity,  is  called  an  arithmetical  pro- 
gressimu 

Thus,  the  natural  numbers  written  in  their  order,  (as  1,  2,  3, 
4,  5,  6,  7,  8,  9,  10,  &c.)  form  an  arithmetical  progression, 
because  they  constantly  increase  by  unity ;  and  the  series  25, 
22,  19,  16,  13,  10,  7,  4,  1,  &c.  is  also  such  a  progression,  since 
the  numbers  constantly  decrease  by  3. 

362.  The  number,  or  quantity,  by  which  the  terms  of  aii 
arithmetical  progression  become  greate^or  less,  is  called  the  dif- 
ference.    So  that  when  the  first  term  and  the  difference  are 

given,  we  may  continue  the  arithmetical  progression  to  any 
length. 

For  example,  let  the  first  term  =  2,  and  the  difference  =  3, 
and  we  shall  have  the  fojjowing  increasing  progression  :  2,  5, 
8,  11,  14,  17,  20,  23,  26,  29,  &c.  in  which  each  term  is  found, 
by  adding  the  difference  to  the  preceding  term.     ^ 

363.  It  is  usual  to  write  the  natural  numbers,  1,  2,  3,  4,  5,  &c. 
above  the  terms  of  ,<^uch  an  arithmetical  progression,  in  order 
that  we  may  immediately  perceive  the  rank  which  any  term 
holds  in  the  progression.     T'iiese  numbers  written  above  the 


Chap.  S.  Of  Compotmd  Quantities,  117 

terms,  may  be  called  indices ;  and  the  above  example  is  written 

as  follows : 

Iiidices,  12345678910 

Jrithm.Prog.    2,    5,     8,   11,   14,  17,  20,  23,  26,  29,  &c. 

where  we  see  that  29  is  the  tenth  term. 

364.  Let  a  be  the  first  term,  and  d  the  difference,  the  arith- 
metical  progression  will  go  on  in  the  following  order  : 

12  3  4  5  6  7 

a,  a  +  d,  a  -^Qd,  a  +  Sd,  a  +  4d,  a  -f  Sd,  a  +  6d,  &c, 
whence  it  appears,  that  any  term  of  the  progression  might  be 
easily  found,  without  the  necessity  of  finding  all  the  preceding 
ones,  by  means  only  of  the  first  term  a  and  the  difference  d. 
For  example,  the  tenth  term  will  be  =  a  +  9d,  the  hundredth 
term  =  a  +  99(/,  and,  generally,  the  term  n  will  be  =  a  -f. 

(71—  \y. 

365.  When  we  stop  at  any  point  of  the  progression,  it  is  of 
importance  to  attend  to  the  first  and  the  last  term,  since  the 
index  of  the  last  will  represent  the  number  of  terms,  /f,  there- 
fore, the  first  term  =  a,  the  difference  =  d,  and  the  number  of  terms 

=  n,  we  shall  have  the  last  term  =  a  -f  (n  —  1)  d,  which  is  con- 
sequently found  hj  mtdtiplying  the  difference  by  the  mimber  of  terms 
minus  one,  and  adding  the  first  term  to  that  product.  Suppose,  for 
example,  in  an  arithmetical  progression  of  a  hundred  terms, 
the  first  term  is  =  4,  and  the  difference  =  3  ;  then  the  last  term 
will  be  =  99  X  3  -I-  4  =  301. 

366.  When  we  know  the  first  term  a  and  the  last  a,  with  the 
number  of  terms  n,  we  can  find  the  difference  d.  For,  since 
the  last  term  »  =  a  -f  (?i  —  ±)d,  if  we  subtract  a  from  both  sides, 
we  obtain  z  —  a  =  (?i  —  1)  d.  So  that  by  subtracting  the  first 
term  from  the  last,  we  have  the  product  of  the  difference  multi- 
plied by  the  number  of  terms  mimis  1.  \^'e  have,  therefore, 
only  to  divide  «;  —  ahy  n  —  1  to  obtain  the  required  value  of 

the  difference  d,  which  will  be  =  ^^.  This  result  furnishes  the 

w— 1 

following  rule  :     Subtract  the  first  term  from  the  last,  divide  tJie 

remainder  by  the  number  of  terms  minus  1,  and  the  quotient  will 

be  the  difference :  by  means  of  which  we  may  write  the  whole 

progression. 


118  '  Algebra.  Sect.  3. 

ser.  Suppose,  for  example,  that  we  have  an  arithmetical 
progjessioji  of  nine  terms,  whose  first  is  =  2,  and  last  =  26, 
and  that  it  is  required  to  find  the  difference.     We  must  subtract 
the  first  term  2  from  the  last  26,  and  divide  the  remainder, 
which  is  24,  by  9  —  1,  that  is  by  8  ;  the  quotient  3  will  be  equal 
to  the  difference  required,  and  the  wliole  progression  will  be  :   " 
123456789 
2,         5,         8,      11,       14,       17,      20,      23,      26. 
To  give  another  example,  let  us  suppose  that  the  first  term 
=  1,  the  last  =  2,  the  number  of  terms  =  10,  and  that  the  arith- 
metical progression,  answering  to  these  suppositions,  is  requir- 

^ I        I 

ed ;  we  shall  immediately  have  for  the  difference, =  — , 

10—1       y 

and  thence  conclude,  that  the  progression  is  : 

123456789  10 

1,*    1|,      1|,      1|,       1|,      1|,       1|,      IJ,       If,        2. 
Another  example.    Let  the  first  term  =  2 J,  the  last  =  12|, 

and  the  number  of  terms  =  7  ;   the  difference  will  be  -^IZISJ' 

7—1 

101.        61       ,25         ,  .,    ,, 

=  — 1.  =  —  =  1— ,  and  consequently  the  progression  : 

O  Ov)  OD 

12  3  4  5  6  7 

Ol  41  '^'^^.        7S  Q"'  1029  lOl 

-^■S'  %?"'  ^T¥'        ^T2'  ^T9         ^^3  6-'        -^^Sf 

268.  If  now  the  first  term  a,  the  last  term  «,  and  the  differ- 
ence d,  aie  given,  we  may  from  them  iind  the  number  of  terms 

11.    For,  since  a —  a  =  (?i  —  1)  d,  by  dividing  the  two  sides 

~^    ft 
by  d,  we  have  — j-  =  n  —  1.     Now,  n  being  greater  by  1 
a 

than  n  —  1,  we  have  n  ^  — j-  -f  1 ;    consequently,   the  number 

of  terms  is  found  by  dividing  the  difference  between  the  first  and  the 
last  term,  or  z  —  a,  btj  the  difference  of  the  progression,  and  adding 

unity  to  the  quotientf  — j-  • 

For  example,  let  the  first  term  =  4,  the  last  =  100,  and  the 

difference  =  12,  the  number  of  terms  will  be  — ~   +1  =  9; 

and  these  nine  terms  will  be, 

123456789 

4,       16,     28,      40,      52,      64,      76,      88,     100. 


Chap.  S,  Of  Compound  ((tiantities,  119 

If  the  first  term  =  2,  the  last  =  6,  and  difference  =  1^,  the 
4 
number  of  terms  will  be  —  -f-  1  =  4 ;    and  these  four  terms 

will  be, 

1         £         3         4 
2,      34,      4f,        6. 
Again,  let  the  first  term  =  3^,  the  last  =  7|,  and  the  difier- 

7*— Si 
ence  =  1|,  the  number  of  terms  will  be  =     ^       ^  +1=45 

H 
which  are, 

*^3'   ^T'   "9"'    'U"' 

369.  It  must  be  observed  however,  that  as  the  number  of  terms 
is  necessarily  an  integer,  if  we  had  not  obtained  such  a  number 
for  w,  in  the  examples  of  tlie  preceding  article,  the  questions 
would  have  been  absurd. 

Whenever  we  do  not  obtain  an  integer  number  for  the  value 

of  —7-9  it  will  be  impossible  to  resolve  the  question ;  and  con- 

a 

sequently,  in  order  that  questions  of  this  kind  may  be  possible, 
55  —  a  must  be  divisible  by  d, 

370.  From  what  has  been  said,  it  may  be  concluded,  that  we 
have  always  four  quantities,  or  things,  to  consider  in  an  arith- 
metical progression ; 

I.  The  first  term  a, 
II.  The  last  term  «. 

III.  The  difference  d. 

IV.  The  number  of  terms  n. 

And  the  relations  of  these  quantities  to  each  other  are  such,  that 
if  we  know  three  of  them,  we  are  able  to  determine  the  fourth ; 
for, 

I.  If  a,  d,  and  n  are  known,  we  have  z  =  a  -f-  (n  —  IJd. 
II.  If  z,  d,  and  n  are  knowiif  we  have  a  =  z  —  (n  —  l)d. 

III.  If  a,  z,  and  n  are  known,  we  have  d  =  -^,. 

n — 1 

IV.  If  a,  z,  a7i(kd  are  known,  we  have  n  =  -^  -f  1. 


12«  '  ^..  ,   jSgebra,  Sect.  Si 

CHAPTER  IV. 

Of  the  Summation  of  Arithmetical  Progressions. 

S71.  It  is  often  necessary  also  to  find  the  sum  of  an  arith- 
metical progression.  This  might  be  done  by  adding  all  the 
terms  together ;  but  as  the  addition  would  be  very  tedious,  when 
the  progression  consisted  of  a  great  number  of  terms,  a  rule  has 
been  devised,  by  which  the  sum  may  be  more  readily  obtained. 

372.  We  shall  first  consider  a  particular  given  progression, 
such  that  the  first  term  =  2,  the  difference  =  3,  the  last  term 
=  29,  and  the  number  of  terras  =  10  ^ 

123456789  10 

2,  5,  8,  11,  14,  17,  20,  23,  26,  29. 
We  see,  in  this  progression,  that  the  sum  of  the  first  and  the 
last  term  =  31 ;  the  sum  of  the  second  and  the  last  but  one 
=  31 5  the  sum  of  the  third  arid  the  last  but  two  =  31,  and  so 
on;  and  thence  w^e  conclude,  that  the  sum  of  any  two  terms 
equally  distant,  the  one  from  the  first,  and  the  other  from  the  last 
term,  is  always  equal  to  the  sum  of  the  first  and^tlic  last  term. 

373.  The  reason  of  this  may  be  easily  traced,  ^or,  if  we  sup- 
pose the  first  =  a,  the  last  =  »,  and  the  difference  =  d,  the  sum 
of  the  first  and  the  last  term  is  =  a  -f  a ;  and  the  second  term 
being  =  a  -}-  d,  and  the  last  but  one  =  «  —  d,  the  sum  of  these 
two  terms  is  also  =  a  -f  ®.  Further,  the  third  term  being  a  + 
2(if  and  the  last  but  two  =  «  —  2d,  it  is  evident  that  these  two 
terms  also,  when  added  together,  make  a  -f  ».  The  demon- 
stration may  be  easily  extended  to  all  the  rest. 

374.  To  determine,  therefore,  the  sum  of  the  progression 
proposed,  let  us  write  the  same  progression  term  by  term, 
inverted,  and  add  the  corresponding  terms  together,  as  follows  : 

2  -f.  5  +  8  4-  1 1  -f.  14  4-  17  -f  20  -f  23  -f.  26  -f  29 
29  4- 26 -f- 23  +  20-f  17  -f-  14  -h  11  +   8  +  54-2. 

31  4-31 -f-31  -f.  31  4-31  4-  31  4-  31  -f  31  4.  31  4.  31 
This  series  of  equal  terms  is  evidently  equal  to  twice  the  sum 
of  the  given  progression  ;  now  the  number  of  these  equal  terms 
is  10,  as  in  the  progression,  and  their  sum,  consequently,  =  10 


Chap.  4.  OJ  Compound  Quantities,  121 

X  31  =  310.  So  that,  since  this  sum  is  twice  the  sum  of  the 
arithmetical  progression,  the  sum  required  must  be  =  155. 

375.  If  we  proceed,  in  the  same  manner,  with  respect  to  any 
arithmetical  progression,  the  first  term  of  which  is  =  a,  the  last 
=  %,  and  the  number  of  terms  =  n  ;  writing,  under  the  given  - 
progression,  the  same  progression  inverted,  and  adding  term  to 
term,  we  sliall  have  a  series  of  n  terms,  each  of  which  will  be 
=  a  -f- »  ;  the  sum  of  tliis  series  will  consequently  be  =  w  (a  +«), 
and  it  will  be  twice  the  sum  of  the  proposed  arithmetical  pro- 
gression ;  wliich  therefore  will  be  =    ^        ' . 

376.  This  result  furnishes  an  easy  method  of  finding  the  sum 
of  any  arithmetical  progression ;  and  may  be  reduced  to  the 
following  rule  : 

Multiply  the  siim  of  the  first  and  the  last  term  by  the  mimber  of 
tenns,  and  half  the  product  will  be  the  sum  of  the  whole  progres- 
sion. 

Or,  which  amounts  to  the  same,  multiply  the  sum  of  the  first 
and  the  last  term  by  half  the  number  of  terms. 

Or,  multiply  half  the  sum  of  the  first  and  the  last  term  by  the 
whole  number  of  terms.  Both  these  enunciations  of  the  rule 
will  give  the  sum  of  the  progression. 

377.  It  may  be  proper  to  illustrate  this  rule  by  some  exam- 
ples. 

First,  let  it  be  required  to  find  the  sum  of  the  progression  of 
the  natural  numbers,  1,-2,  3,  &c.  to  100.     This  will  be,  by  the 

first  rule,  =  ^^^^^^^  =  50  x  101  =  5050. 

2 

If  it  were  required  to  tell  how  many  strokes  a  clock  strikes 
in  twelve  hours ;  we  must  add  together  the  numbers  1,  2,  3,  as 

12  X  1  *? 

far  as  12  ;  now  this  sum  is  found  immediately  =  ■■■       .  z=  6  x 

13  =  78.  If  we  wished  to  know  the  sum  of  the  same  progres- 
sion continued  to  1000,  we  should  find  it  to  be  500500 ;  and  the 
sum  of  this  progression  continued  to  10000,  would  be  50005000, 

378.  Another  question.  A  person  buys  a  horse,  on  condition 
that  for  the  first  nail  he  shall  pay  5  halfpence,  for  the  second  8, 
for  the  third  11,  and  so  on,  always  increasing  3  halfpence  more 

16 


ia2  Mgehra.  Sect.  S» 

for  each  following  one  ;  the  horse  having  32  nails,  it  is  required 
to  tell  how  much  he  will  cost  the  purchaser  ? 

In  this  question,  it  is  required  to  find  the  sum  of  an  arith- 
metical progression,  the  first  term  of  which  is  5,  the  diiference 
=  3,  and  the  number  of  ter»ns  =  32.  We  must  therefore  begin 
by  determining  the  last  term  ;  we  find  it  (by  the  rule  in  articles 
365  and  370)  =  5  +  31  x   3  =  98.     After  which  tlie  sum  re- 

103  X  32 
quired  is  easily  found  =  ^^^  =  103  x  16  ;  whence  we  con- 

elude,  that  the  horse  costs  1648  halfpence,  or  3^  8s.  8d, 

379.  Generally,  let  the  first  term  be  =  a,  the  difference  =  d^ 
and  the  number  of  terms  =  n  ;  and  let  it  be  required  to  find,  hy 
means  of  these  data,  the  sum  of  the  whole  progression.  As  the 
last  term  must  be  =  a  +  («  —  l)d,  the  sum  of  the  first  and  last 
will  be  =  2a  -f-  (n  —  l)d.  Multiplying  this  sum  by  the  number 
of  terms  n,  we  have  ^na  -f-  n  (ji  —  1)  fi ;  the  sum   required 

therefore  will  be  =  na  +  ^H^      )    ^ 

^  2 

This  formula,  if  applied  to  the  preceding  example,  or  to  a  =  5, 
d  =  3,  and  n  =  32,  gives  5  x  32  -f-  -~^l—±-  =  160  +  1488  = 

1648  ;  the  same  sum  that  we  obtained  before. 

380.  If  it  be  required  to  add  together  all  the  natural  numbers 

from  1  to  n,  we  have,  for  finding  this  sum,  the  first  term  =  1, 

the  last  term  =  n,  and  the  number  of  terms  =  n ;  wiierefore  the 

,,  •      1  •  nn-\-n       n{n-\-V\ 

the  sum  required  is  =  — ;-i—  =  --^ — ■ — ^. 

If  we  make  n  =  1766,;  the  sum  of  all  the  numbers,  from  1  to 
1766,  will  be  =  883  X  1767  =  1560261. 

381.  Let  the  progression  of  uneven  numbers  he  proposed,  1,  3,  5, 
7,  ^'c.  continued  to  n  terms,  and  let  the  sum  of  it  be  required  : 

Here  the  first  terni  is  =  1,  the  difference  =  2,  the  number  of 
terms  =  n  •  the  last  term  will  therefore  be  =  1  -f  (n  —  1)  2  = 
£re  —  1,  and  consequently  the  sum  required  =  nn. 

The  whole  therefore  consists  in  multiplying  the  number  of 
terms  by  itself.  So  that  whatever  numher  of  terms  of  this  pro- 
gression we  add  together ,  the  sum  will  he  always  a  square,  7iamely9 
the  square  of  the  numher  of  terms.  This  we  shall  exemplify  as 
follows ; 


Chap.  4.  Of  Comjjound  Quantities,  12S 

Indices,  1  2  3  4  5  6  7  8  9  10  &c. 
Progress,  1,  3,  5,  7,  9,  11,  13,  15,  17,  19,  &c. 
Sum.  1,      4,      9,     16,    25,    36,  49,    64,    81,    100,  &c. 

382.  Let  the  first  term  be  =  1,  the  difference  =  3>  and  the 
number  of  terms  =  n ;  we  shall  have  the  progression  1,  4,  7, 
10,  6cc.  the  last  term  of  which  will  bel  +  (?i  —  l)3=3ii^-2; 
wherefore  the  sum  of  the  first  and  the  last  term  =  3?*  —  1,  and 

consequently,  the  sum  of  this  progression  =  — ^^ '-  = • 

If  we  suppose  n  =  20,  the  sum  will  be  =  10  x  59  =  590. 

383.  Again,  let  the  first  term  =  1,  the  difference  =  d,  and 
the  number  of  terms  =  n ;  then  the  last  term  will  be  =  1  + 
(n  —  l)d.  Adding  the  first,  we  have  2  +  (w  —  l)  d,  and  mul- 
tiplying by  the  number  of  terms,  we  have  2ti  +  n  (n  —  i)  d; 

whence  we  deduce  the  sum  of  the  progression  =  ti  -f-  ^^^      /  . 

We  subjoin  the  following  small  table  : 

If  rf  =  1,  the  sum  is  =  ?i  +  —^ — ^    =   -~- 

2  2 

a  =  2,  =  w  -I ^ — ^  =  nu 

a  =  4,  =  «  -( ^^ '  =  2nn  —  n 

J       ^  5n(n — \)        Snn — 3n 

^22 

d  =  6,  =  71  4 ^  '     =  37171  27« 

^  2 

,       _,                                      7n(n — 1\         771W — 5n 
dz=7,  =  71  H i^ )   =  

rf  =  8,  z=  11  4 ^ ^  =  47171  —  371 

2 

(1  =  9,  =  71  +    — ii ^  =    

^2  2 

rf  =  10,  =n  -\ — ^^^      ^  =  Snn  —  4n 


J54  Algebra,  Sect.  8, 


"O 


CHAPTER  V. 

Of  Geometrical  Ratio, 

384.  The  geometrical  ratio  of  two  numbers  is  found  by  resolv- 
ing tbe  question,  how  many  times  is  one  of  those  numbers 
greater  tlian  ttie  other  ?  This  is  done  by  dividing  one  by  the 
other ;  and  tlie  quotient,  therefore,  expresses  the  ratio  required. 

385.  We  have  here  three  things  to  consider  ;  1st,  the  first  of 
the  two  given  numbers,  which  is  called  the  antecedent ;  2dly, 
the  other  number,  wliich  is  called  the  consequent;  3dly,  the 
ratio  of  the  two  numbers,  or  the  quotient  arising  from  the  divis- 
ion of  the  antecedent  by  the  consequent.  For  example,  if  the 
relation  of  the  numbers  18  and  12  be  required,  18  is  the  antece- 
dent, 1^  is  the  consequent,  and  the  ratio  will  be  \^  =  1| ; 
whence  w^e  see,  that  the  antecedent  contains  the  consequent  once 
and  a  half. 

386.  It  is  usual  to  represent  geometrical  relation  by  two 
points,  placed  one  above  the  other,  between  the  antecedent  and 
the  consequent.  Thus  a  :  h  means  the  geometrical  relation  of 
these  two  numbers,  or  the  ratio  of  h  to  a. 

We  have  already  remarked,  that  this  sign  is  employed  to 
represent  division,  and  for  this  reason  we  make  use  of  it  here ; 
because,  in  order  to  know  the  ratio,  we  must  divide  a  by  &. 
The  relation,  expressed  by  this  sign,  is  read  simply,  a  is  to  &. 

387.  Relation  therefore  is  expressed  by  a  fraction,  whose 
numerator  is  the  antecedent,  and  whose  denominator  is  the  con- 
sequent. Perspicuity  requires  that  this  fraction  should  be 
always  reduced  to  its  lowest  terms  ;  which  is  done,  as  we  have 
already  shewn,  by  dividing  both  the  numerator  and  denominator 
by  their  greatest  common  divisor.  Thus,  the  fraction  i|  be- 
comes |,  by  dividing  both  terms  by  6. 

388.  So  that  relations  only  differ  according  as  their  ratios 
are  different ;  and  there  are  as  many  different  kinds  of  geometri- 
cal relations  as  we  can  conceive  different  ratios. 

The  first  kind  is  undoubtedly  that  in  which  the  ratio  becomes 
unity  ;  this  case  happens  when  the  two  numbers  are  equal,  as 
in  3  :  3  ;  4  :  4  ;  a  :  a  ;  the  ratio  is  here  1,  and  for  this  I'eason  we 
call  it  the  relation  of  equality. 


Chap.  5.  OfCompmmd  Quantities,  125 

Next  follow  those  relations  in  which  the  ratio  is  another  whole 
number  ;  in  4  :  2  the  ratio  is  2,  and  is  called  dmible  ratio  ;  in  12  ^ 
4  the  ratio  is  3,  and  is  called  triple  ratio  ;  in  24  :  6  the  ratio  is  4, 
and  is  called  quadruple  ratio,  &c. 

We  may  next  consider  those  relations  whose  ratios  are  expres- 
sed by  fractions,  as  12 :  9,  where  the  ratio  is  4  or  14  ;  18  :  27, 
were  the  ratio  is  |,  &c.  We  may  also  distinj^uish  tliose  rela- 
tions in  which  the  consequent  contains  exactly  twice,  thrice,  &c. 
the  antecedent ;  such  are  the  relations  6 :  12,  5  :  15,  &c.  the  ra- 
tio of  which  some  call,  subdMple,  suhtriplef  &c.  ratios. 

Further,  we  call  that  ratio  rational,  which  is  an  expressible 
number ;  the  antecedent  and  consequent  being  integei^,  as  in 
11  :  7,  8  :  15,  &c.  and  we  call  that  an  irrational  or  surd  ratio, 
which  can  neither  be  exactly  expressed  by  integers,  nor  by  frac- 
tions, as  in  v/  5  :  8,  4  :  V  3. 

389.  Let  a  be  the  antecedent,  h  the  consequent,  and  d  the  ra- 
tio, we  know  already  that  a  and  b  being  given,  we  find  d  =  4. 

b 

If  the  consequent  b  were  given  with  the  ratio,  we  should  find 
the  antecedent  a  =  bd,  because  bd  divided  by  b  gives  d.  Lastly, 
when  the  antecedent  a  is  given,  and  the  ratio  d,  we  find  the" 

consequent  b  =  —  ;  for,  dividing  the  antecedent  a  by  the  conse- 

(i 

quent  -^,  we  obtain  the  quotient  d,  that  is  to  say,  the  ratio. 

390.  Every  relation  a  ;  b  remains  the  same,  though  we  multi- 
ply, or  divide  the  antecedent  and  consequent  by  the  same  num- 
ber, because  the  ratio  is  the  same.    Let  d  be  the  ratio  of  a :  &^ 

we  have  dz=z  —;  now  the  ratio  of  the  relation  na  :  nb  is  also  -- 
0  b 

n        h  ft 

=  (Z,  and  that  of  the  relation  --  :  —  is  likewise  —  =  d. 

n     n  b 

391.  When  a  ratio  has  been  reduced  to  its  lowest  terms,  it  is 
easy  to  perceive  and  enunciate  the  relation.  For  example,  when 

CL  7) 

the  ratio  -7  has  been  reduced  to  the  fraction  — ,  we  say  a:  b=: 
b  q 

p  :  q,  a:b::p:  q,  which  is  read,  a  is  to  6  as  p  is  to  q.      Thus, 

the  ratio  of  the  relation  6  : 3  being  |,  or  2,  we  say  6:3=2:1. 


126  Mgebra.  Sect.  3. 

We  have  likewise  18  :  12  =  3  :  2,  and  24  :  18  =  4  :  5,  and  30  :  45 
s=  2  :  3,  &c.  But  if  the  ratio  cannot  he  abridged,  the  relation 
will  not  become  more  evident ;  we  do  not  simplify  the  relation 
by  saying  9:7  =  9:7. 

392.  On  the  other  hand,  we  may  sometimes  change  the  rela- 
tion of  two  very  great  numbers  into  one  that  shall  be  more 
simple  and  evident,  by  reducing  both  to  their  lowest  terms.  For 
example,  we  can  say  28844  :  14422  =  2:1;  or,  10566  :  7044 
=  3:2;  or,  57600  :  25200  =16:7. 

393.  In  order,  therefore,  to  express  any  relation  in  the  clear- 
est manner,  it  is  necessary  to  reduce  it  to  the  smallest  possible 
numbers.  This  is  easily  done,  by  dividing  the  two  terms  of  the 
relation  by  their  greatest  common  divisor.  For  example,  to 
reduce  the  relation  57600  :  25200  to  that  of  16  :  7,  we  have  only 
to  perform  the  single  operation  of  dividing  the  numbers  576  and 
252  by  36,  which  is  their  greatest  common  divisor. 

394.  It  is  important,  therefore,  to  know  how  to  find  the  great- 
est common  divisor  of  two  given  numbers ;  but  this  requires  a 
rule,  which  we  shall  explain  in  the  following  chapter. 


CHAPTER  VI. 

Of  the  greatest  Common  Divisor  of  two  given  numbers^ 

395.  There  are  some  numbers  which  have  no  other  common 
divisor  than  unity,  and  when  the  numerator  and  denominator 
of  a  fraction  are  of  this  nature,  it  cannot  be  reduced  to  a  more 
convenient  form.  The  two  numbers  48  and  35,  for  example, 
have  no  common  divisor,  though  each  has  its  own  divisors. 
For  this  reason  we  cannot  express  the  relation  48  :  35  more 
simply,  because  the  division  of  two  numbers  by  1  does  not 
diminish  them. 

396.  But  when  the  two  numbers  have  a  common  divisor,  it  is 
found  by  the  following  rule : 

Divide  the  greater  of  the  two  numbers  by  the  less  ;  nextf  divide 
the  jjreceding  divisor  by  the  remaimler ;  what  remains  in  this 
second  division  will  afterwards  become  a  divisor  for  a  third  divis- 
ion, in  which  the  remainder  of  the  preceding  divisor  will  be  the 


Chap.  6,  OJ  Compmind  ^uantities^  127 

dividend.  We  must  continue  this  operation,  till  we  arrive  at  a 
division  that  leaves  no  remainder  ;  the  divisor  of  this  division,  and 
consequently  the  last  divisor,  will  be  the  greatest  common  divisor  of 
the  two  given  numbers. 

See  this  operation  for  the  two  numbers  576  and  252. 
252)  576  (2 
504 

•^  72)  252  (3 

216 

36)  72  (2 

72 

0. 
So  that,  in  this  instance,  the  greatest  common  divisor  is  36, 

397.  It  will  be  proper  to  illustrate  this  rule  by  some  other 
examples.  Let  the  greatest  common  divisor  of  the  numbers 
504  and  312  be  required. 

312)  504  (1 
312 

192)  312  (1 
192 

120)  192(1 
120 

72)  120  (1 

72 

48)72  (1 
48 

24)  48  (2 
48 

0. 
So  that  24  is  the  greatest  common  divisor,  and  consequently 
the  relation  504  :  312  is  reduced  to  the  form  21  :  13. 

398.  Let  the  relation  625  :  529  be  giMJn,  and  the  greatest 
common  divisor  of  those  two  numbers  be  required. 


'2«  Mgebra.  Sect.  S. 


529) 

625 
529 

(1 

96] 

1529 
480 

(5 

49] 

>96 
49 

IT) 

(1 

49(1 

47 

2)  47  (2S 
46 

1)2(2 
2 

0. 
Wherefore  1  is,  in  this  case,  the  greatest  common  divisor, 
nd  consequently  we  cannot  express  the  relation  625  :  529  by 
ess  numbers,  nor  reduce  it  to  less  terms, 

399.  It  may  be  proper,  in  this  place,  to  give  a  demonstration 
of  the  rule.  In  order  to  this,  let  a  be  the  greater  and  b  the  less 
of  the  given  numbers  ;  and  let  d  be  one  of  their  common  divisors  ; 
it  is  evident  that  a  and  b  being  divisible  by  d,  we  may  also 
divide  the  quantities  a  —  b.  a  —  2&,  a  —  36,  and,  in  general 
U'—nbhyd. 

400.  The  converse  is  no  less  true  :  that  is  to  say,  if  the  num- 
bers b  and  a  —  nb  are  divisible  by  d,  the  number  a  will  also  bo 
divisible  by  d.  For  nb  being  divisible  by  d,  we  could  not  divide 
a  —  nb  by  d,  if  a  were  not  also  divisible  by  d, 

401.  We  observe  further,  that  if  d  be  the  greatest  common 
divisor  of  two  numbers,  b  and  a  —  ?i&,  it  will  also  be  the  great- 
est common  divisor  of  the  two  numbers  a  and  b.  Since,  if  a 
greater  common  divisor  could  be  found  than  d,  for  these  num- 
bers,, a  and  6,  that  number  would  also  be  a  common  divisor  of  & 
and  a  —  nb;  and,  consequently,  d  would  not  be  the  greatest 
common  divisor  of  these  two  numbers.  Now  we  have  supposed 
d  the  greatest  divisor  common  to  b  and  a  —  nb  ;  wherefore  d 
must  also  be  the  greatest  common  divisor  of  a  and  b. 


Chap.  6.  OJ  Compound  l^iantities,  129 

402.  These  three  things  being  laid  down,  let  us  divide, 
according  to  the  rule,  the  greater  number  a  by  the  less  b ; 
and  let  us  suppose  the  quotient  =  n ;  the  remainder  will  be 
a  —  nbf  which  must  be  less  than  &.  Now,  tbis  remainder  a  —  nh 
having  the  same  greatest  common  divisor  with  6,  as  the  given 
iiumbei*s  a  and  b,  we  have  only  to  repeat  the  division,  dividing 
the  preceding  divisor  b  by  the  remainder  a  —  nb;  the  new 
remainder,  which  we  obtain,  will  still-have,  with  the  preceding 
divisjr,  the  same  greatest  common  divisor,  and  so  on. 

403.  We  proceed  in  the  same  manner,  till  we  arrive  at  a 
division  without  a  remainder ;  that  is,  in  which  the  remainder 
is  nothing.  Let  p  be  the  last  divisor,  contained  exactly  a  cer- 
tain number  of  times  in  its  dividend  ;  this  dividend  will  there- 
lore  be  divisible  by  p,  and  will  have  the  form  mp ;  so  that  the 
numbers  ];,  and  tn_p,  are  both  divisible  by  p  ;  and  it  is  certain, 
that  they  have  no  greater  common  divisor,  because  no  number 
can  actually  be  divided  by  a  number  greater  than  itself.  Con- 
sequently, tliis  last  divisor  is  also  tiie  greatest  common  divisor 
of  the  given  numbers  a  and  b,  and  the  rule,  w  hich  w^e  laid  down, 
is  demonstrated. 

404.  We  may  give  another  example  of  the  same  rule,  requir- 
ing the  greatest  common  divisor  of  the  numbers  1728  and  2304, 
The  operation  is  as  follows  : 

1728)  2304  (1 
1723 

576)  1728  (3 
1728 

0. 
From  this  it  follows,  that  576  is  the  greatest  common  divisor, 
and  that  the  relation  1728  :  2304  is  reduced  to  3  :  4  5  that  is  to 
say,  1728  is  to  2304  the  same  as  3  is  to  4. 


17 


180  Mgehra,  Sect.  3. 

CHAPTER  VII. 

Of  Geometrical  Proportions. 

405.  Two  geometrical  relations  are  equal,  when  their  ratios 
are  equal.  Tliis  equality  of  two  relations  is  called  a  geometrical 
proportion  ;  and  we  write  for  example,  a  :  6  =  c  :  (f ,  or  a  :  6  :  :  c 
'  d,  to  indicate  that  the  relation  a  :  6  is  equal  to  the  relation 
c.  d;  but  this  is  more  simply  expressed  by  saying,  a  is  to  &  as 
c  to  d.  The  following  is  such  a  proportion,  8:4=12:6;  for 
the  ratio  of  the  relation  8  :  4  is  4,  and  this  is  also  the  ratio  of 
the  relation  12  :  6. 

406.  So  that  a  :  b  =  c  :  d  being  a  geometrical  proportion,  the 

ratio  must  be  the  same  on  both  sides,  and  ~  =  -j  j    and,  recip- 

b       a 

rocally,  if  the  fractions  —  and  ~  are  equal,  we  have  aib  wed, 

407.  A  geometrical  proportion  consists  therefore  of  four  terms, 
such,  that  the  first,  divided  by  the  second,  gives  the  same  quo- 
tient as  the  third  divided  by  the  fourth.  Hence  we  deduce  an 
important  property,  common  to  all  geometrical  proportion, 
which  is,  that  the  product  of  the  first  and  the  last  term  is  always 
equal  to  the  product  of  the  second  and  third  ;  or,  more  simply,  that 
the  product  of  the  extremes  is  equal  to  the  product  of  the  means, 

408.  In  order  to  demonstrate  this  property,  let  us  take  the 

geometrical  proportion  a  :  6  =  c  :  J,  so  that  —=--,.  If  we  mul- 

bc 
tiply  both  these  fractions  by  b,  we  obtain  a  =  -75  and  multiply- 

ing  both  sides  further  by  d,  we  have  ad  =  be.  Now  ad  is  the 
product  of  the  extreme  terms,  be  is  that  of  the  means,  and  these 
two  products  are  found  to  be  equal. 

409.  RedprocaUy,  if  the  four  numbers  a,  b,  c,  d,  are  such,  that 
the  product  of  the  two  extremes  a  and  d  is  equal  to  the  product  of 
the  two  means  b  and  c,  we  are  certain  that  they  form  a  geometri- 
cal proportion.    For,  since  ad  =  be,  we  have  only  to  divide  both 

sides  by  bd,  which  gives  us  -r^  =  t-.?  ov  -t  =  -.,  and  conse- 
quently a  :  b  =  c  :  d. 


Chap.  7.  Of  Compound  Quantities,  131 

410.  Thefmir  terms  of  a  geometrical  proportion,  as  a :  b  =  c :  d, 
may  be  transposed  in  different  ways,  without  destroying  the  pro» 
portion.  For  the  rule  being  always,  that  the  product  of  the  eo?- 
tremes  is  equal  to  the  product  of  the  means  or  ad  =  be,  we  may  say  : 
1«^  b  :  a  =  d  :  c  ;  ^'"y-  a  :  c  =  b  :  d ;  3«"y-  d  :  b  =  c  :  a ;  4«Jy- 
d  :  c  =  b  :  a. 

411.  Beside  these  four  geometrical  proportions,  we  may  de- 
duce some  others  from  the  same  proportion,  a  i  b  =  c  :  d.  We 
may  say,  the  first  term,  plus  the  second,  is  to  the  first,  as  the  third 
4-  the  fourth  is  the  third  ;  that  is,  a  +  b  :  a  =  c  +  d  :  c. 

We  may  further  say  ;  the  first  —  the  second  is  to  the  first  as 
the  third  —  the  fourth  is  to  the  third,  or  a  —  b  :  a  =  c  —  die. 

For,  if  we  take  the  product  of  the  extremes  and  the  means, 
we  have  ac  —  be  =  ac  —  ad,  which  evidently  leads  to  the  equal- 
ity ad  =  be. 

Lastly,  it  is  easy  to  demonstrate,  that  a  +  b:bz=:c-^d:d; 
and  that  a  —  b  :  b  =  c  —  d  :  d. 

412.  All  the  proportions  which  we  have  deduced  from  a  :  6  = 
c  ;  d,  may  be  represented,  generally,  as  follows  : 

ma  -f-  nb  :  pa  -{-  qb  =  mc  -^  nd  :  pc  -f  qd. 
For  the  product  of  tlie  extreme  terms  is  mpac  +  npbc  +  mqad 
+  nqbd  ;  which,  since  ad  =  be,  becomes  mpac  +  npbc  +  mqbc  -f 
nqbd.  Further,  the  product  of  the  mean  terms  is  mpac  -f-  viqbc 
+  npad  -f  nqbd  ;  or,  since  ad  =  be,  it  is  mpac  -f  mqbc  +  npbc  -f 
nqbd  ;  so  that  the  two  products  are  equal. 

413.  It  is  evident,  therefore,  that  a  geometrical  proportion 
being  given,  for  example,  6  :  3  =  10  :  5,  an  infinite  number  of 
others  may  be  deduced  from  it.     We  shall  give  only  a  few : 

3  :  6  =  5  :  10  ;  6  :  10  =  3  :  5  ;  9  :  6  =  15  :  10  ; 
3:3  =  5:     5  ;   9  :  15  =  3  :  5  ;  9  :  3  =  15  :     5. 

414.  Since,  in  every  geometrical  proportion,  the  product  of 
the  extremes  is  equal  to  the  product  of  the  means,  we  may, 
when  the  three  first  terms  are  known,  find  the  fourth  from  them. 
Let  the  three  first  terms  be  24  :  15  =  40  to  ....  as  the  product 
of  the  means  is  here  600,  the  fourth  term  multiplied  by  the  first, 
that  is  by  24,  must  also  make  600 ;  consequently,  by  dividing 
600  by  24,  the  quotient  25  will  be  the  fourth  term  required,  and 
the  whole  proportion  will  b^  24  :  15  =  40  :  25.    In  general. 


1^  Algebra,  Sect.  8. 

therefore,  if  the  thi*ee  first  terras  are  a  :  6  =  c  :  .  .  .  .  we  put  (i 

for  the  unknown  fourth  letter ;  and  since  ad  =  bcj  we  divide 

be 
both  sides  by  a,  and  have  d  =  —  .     So  that  the  fourth  term  is 

be 
=  — ,  and  is  found  by  multiplying  the  second  term  by  the  third,  and 

dividing  that  product  by  the  first  term, 

415.  This  is  the  foundation  of  the  celebrated  Rule  of  Three  in 
arithmetic ;  for  what  is  required  in  that  rule  ?  We  suppose  three 
numbers  given,  and  seek  a  fourth,  which  may  be  in  geometrical 
proportion  ;  so  that  the  first  may  be  to  the  second,  as  the  third 
is  to  the  fourth. 

416.  Some  particular  circumstances  deserve  attention  here. 
First,  if  in  two  proportions  the  first  and  the  third  terms  arc  the 

same,  ^s  in  a  :  b  —  c  :  d,  and  a  :f=i  c  :  g,  I  say  that  the  two 
second  and  the  two  fourth  terms  will  also  be  in  geometrical  propor- 
tion, and  that  b  :  d  =  f :  g.  For,  the  first  proportion  being 
transformed  into  this,  a  :  c  =  b  i  d,  and  the  second  into  this, 
a  :  c  =zf :  gf  it  follows  that  the  relations  b  :  d  and/:  g  are  equal, 
since  each  of  them  is  equal  to  the  relation  a  :  c.  For  example, 
if  5  ;  100  =  2  :  40,  and  5:15=  2:6,  we  must  have  100  :  40 
=  15  ;  6. 

417.  But  if  the  two  proportions  are  such,  that  the  mean  terms 
are  the  same  in  both,  I  say  that  the  first  terms  will  be  in  an 
inverse  proportion  to  the  fourth  terms.  That  is  to  say,  if  a  :  & 
=zc:  d,  and/:  b  =  c  :  g,  it  follow^s  that  a  :f=  g  :  d.  Let  the 
proportions  be,  for  example,  24  :  8  =  9  :  3,  and  6  :  8  =  9  :  12, 
we  have  24  :  6  =  12  :  S.  The  reason  is  evident ;  the  first  pro- 
portion gives  ad  z=zbc;  the  second  gives  fg  z=bc;  therefore  ad 
:=:fg,  and  a  :f=z  g  :  d,  or  a:g:  :/:  d. 

418.  Two  proportions  being  given,  we  may  always  produce 
a  new  one,  by  sej)arately  multiplying  the  first  term  of  the  one 
by  the  first  term  of  the  other,  the  second  by  the  second,  and  so 
on,  with  respect  to  the  other  terms.  Thus,  the  proportions  a  :  h 
=  c:  d  and  e:fz=g:  h  will  furnish  this,  ae:  bfz=z  eg  :  dh.  For 
the  first  giving  af!  =  be,  and  the  second  giving  eh  =/g",  we  have 
also  adch  =  bcfg.  Now  adeh  is  the  product  of  the  extremes,  and 
bcfg  is  the  product  of  the  means  in  the  new  proportion,*  so  that 
the  tw^o  products  being  equal,  the  pi  oportion  is  true. 


Chap.  8.  Of  Cmnpound  ^tantities.  13S 

419.  Let  the  two  proportions  be,  for  example,  6  :  4  =  15  :  10 
and  9  :  12  =  15  :  20,  their  combination  will  give  the  proportion 
6  X  9  :  4  X  12  =  15  X  15  :  10  X  20, 

or  54  :  48  =  225  :  200, 
or    9  :    8  =      9  :       8. 

420.  We  shall  observe  lastly,  that  if  two  products  are  equal, 
ad  =  be,  we  may  reciprocally  convert  this  equality  into  a  geo- 
metrical proportion  ;  for  we  shall  always  have  one  of  the  factors 
of  the  first  product,  in  the  same  proportion  to  one  of  tlie  factors 
of  the  second  product,  as  the  other  factor  of  tlie  second  product 
is  to  the  other  factor  of  the  first  product ;  that  is,  in  tlie  present 
case,  a:  c  =  b:  d,  or  a  :  b  =  c  :  d.  Let  3x8  =  4  x  6,  and  wc 
may  form  from  it  this  proportion,  8  :  4  =  6  :  3,  or  this,  3:4  = 
6  :  8.  Likewise,  if  3  x  5  =  1  x  15,  we  sliall  have  3:15=1  :  5, 
or  5  :  1  =  15  :  3,  or  3  :  1  =  15  :  5. 


CHAPTER  Vin. 

Observations  an  the  Rules  of  Proportion  and  their  utility, 

421.  This  theory  is  so  useful  in  the  occurrences  of  common 
life,  that  scarcely  any  person  can  do  without  it.  There  is  always 
a  proportion  between  prices  and  commodities  ;  and  when  differ- 
ent kinds  of  money  are  the  subject  of  exchange,  the  whole  con- 
sists in  determining  their  mutual  relations.  The  examples 
furnished  by  these  reflections,  will  be  very  proper  for  illustrating 
the  principles  of  proportion,  and  shewing  their  utility  by  the 
application  of  them. 

422.  If  wc  wished  to  know,  for  example,  the  relation  between 
two  kinds  of  money  ;  suppose  an  old  louis  d'or  and  a  ducat ;  we 
must  first  know  the  value  of  those  pieces,  when  compared  to 
others  of  the  same  kind.  Thus,  an  old  louis  being,  at  Berlin, 
worth  5  rix  dollars*  and  8  drachms,  and  a  ducat  being  worth 
3  rix  dollars,  we  may  reduce  these  two  values  to  one  denomina- 
tion ;  either  to  rix  dollars,  which  gives  the  proportion  1  L  :  1  D 

*  The  rix  dollar  of  Germany  is  valued  at  92  cents  6  mills,  and  a  drachm  is 
one  twenty  fourth  part  of  a  rix  dollaF. 


I 


134  Mgebra.  Sect.  3. 

=  54  R  :  3  R,  or  =  16  :  9 ;  or  to  drachms,  in  which  case  we 
liave  1  L  :  1  D  =  128  ;  72  =  16  :  9.  These  proportions  evi- 
dently give  the  true  relation  of  the  old  louis  to  the  ducat ;  for 
the  equality  of  the  products  of  the  extremes  and  the  means  gives^ 
in  both,  9  louis  =  16  ducats ;  and,  by  means  of  this  comparison, 
we  may  change  any  sum  of  old  louis  into  ducats,  and  vice  versa. 
Suppose  it  were  required  to  tell  how  many  ducats  there  are  in 
1000  old  louis,  we  have  this  rule  of  three.  If  9  louis  are  equal 
to  16  ducats,  what  are  1000  louis  equal  to  ?  The  answer  will 
be  1777^  ducats. 

If,  on  the  contrary,  it  were  required  to  find  how  many  old 
louis  d'or  there  ai^e  in  1000  ducats,  we  have  the  following  pro- 
portion. If  16  ducats  are  equal  to  9  louis;  what  are  1000 
ducats  equal  to  ?    dnswer,  562|  old  louis  d'or. 

423.  Here,  (at  Petersburgh,)  the  value  of  the  ducat  varies, 
and  depends  on  the  course  of  exchange.  This  course  determines 
the  value  of  the  ruble  in  stivers,  or  Dutch  half-pence,  105  of 
which  make  a  ducat. 

So  that  when  the  exchange  is  at  45  stivers,  we  have  this  pro- 
portion, 1  ruble  :  1  ducat  =  45  :  105  =  3:7;  and  hence  this 
equality,  7  rubles  =  3  ducats. 

By  this  we  shall  find  the  value  of  a  ducat  in  rubles ;  for  3 
ducats  :  7  rubles  =  1  ducat  : Jinswer,  2-J  rubles. 

If  the  exchange  were  at  50  stivers,  we  should  have  this  pro- 
portion, 1  ruble  ;  1  ducat  =  50  :  105  =  10  :  21,  which  would 
give  21  rubles  =10  ducats ;  and  we  should  have  1  ducat  =  2-j^ 
rubles.  Lastly,  when  the  exchange  is  at  44  stivers,  we  have  1 
ruble :  1  ducat  =  44  :  105,  and  consequently  1  ducat  =  2iJ 
rubles  =  2  rubles  38-j-^^  copecks.^ 

424.  It  follows  from  this,  that  we  may  also  compare  different 
kinds  of  money,  which  we  have  frequently  occasion  to  do  in  bills 
of  exchange.  Suppose,  for  example,  that  a  person  of  this  place 
has  1000  rubles  to  be  paid  to  him  at  Berlin,  and  that  he  wishes 
to  know  the  value  of  this  sum  in  ducats  at  Berlin. 

The  exchange  is  here  at  47|,  that  is  to  say,  one  ruble  makes 
47^  stivers.  In  Holland,  20  stivers  make  a  florin ;  2J  Dutch 
florins  make  a  Dutch  dollar.    Further,  the  exchange  of  Holland 

•  A  copeck  is  ^ig.  part  of  a  ruble,  as  is  easily  deduced  froift  the  above. 


Chap.  8,  Of  Compmmd  ^antities,  133 

with  Berlin  is  at  142,  that  is  to  say,  for  100  Dutch  dollars,  142 
dollars  are  paid  at  Berlin.  Lastly,  the  ducat  is  worth  3  dollars 
at  Berlin, 

425.  To  resolve  the  questions  proposed,  let  us  proceed  step 
by  step.  Beginning  therefore  with  the  stivei-s,  since  1  ruble  = 
471  stivers,  or  2  rubles  =  95  stivers,  we  shall  have  2  rubles  : 
95  stivers  =  1000  : . . . .  Answer ^  47500  stivers.  If  we  go  fur- 
ther and  say  20  stivers  :  1  florin  =  47500stivers  : ....  we  shall 
have  2375  florins.  Further,  2^  florins  =  1  Dutch  dollar,  or  5 
florins  =  2  Dutch  dollars ;  we  shall  therefore  have  5  florins : 
2  Dutch  dollars  =  2375  florins  : . . . .  Answer,  950  Dutch  dollars. 

Then  taking  the  dollars  of  Berlin,  according  to  the  exchange 
at  142,  we  shall  have  100  Dutch  dollars  :  142  dollars  =  950 : 
to  the  fourth  term,  1349  dollars  of  Berlin.  Let  us,  lastly,  pass 
to  the  ducats,  and  say  3  dollars  :  1  ducat  =  1349  dollars  : . . , . 
Answer,  449|  ducats. 

426.  In  order  to  render  these  calculations  still  more  complete, 
let  us  suppose  that  the  Berlin  banker  refuses,  under  some  pre- 
text or  other,  to  pay  this  sum,  and  to  except  the  bill  of  exchange 
without  five  per  cent,  discount ;  that  is,  paying  only  100  instead 
of  105.  In  that  case,  we  must  make  use  of  the  following  pro- 
portion ;  105  ;  100  =  449|  to  a  fourth  term,  which  is  428|| 
ducats. 

427#  We  have  shewn  that  six  operations  are  necessary,  in 
making  use  of  the  Rule  of  Three ;  but  we  can  greatly  abridge 
those  calculations,  by  a  rule,  which  is  called  the  Ride  of  Reduc- 
tion, To  explain  this  rule,  we  shall  first  consider  the  two 
antecedents  of  each  of  the  six  preceding  operations  : 


I. 

2  rubles 

:  95  stivers. 

IL 

20  stivers 

:  1  Dutch  flor. 

III. 

5  Dutch  flor. 

:  2  Dutch  doll. 

IV. 

100  Dutch  doll. 

:   142  dollars. 

V. 

3  dollars 

.^    :  1  ducat. 

VI. 

105  ducats 

:  100  ducats. 

If  wc  now  look  over  the  preceding  calculatiouvS,  we  shall  ob- 
serve, that  we  have  always  multiplied  the  given  sum  by  the 
second  terms,  and  that  we  have  divided  the  products  by  the 
first  5  it  is  evident  therefore,  that  we  shall  arrive  at  the  same 


136 


Algebra, 


Sect.  3, 


results,  by  multiplying,  at  once,  the  sum  proposed  by  the  pro- 
duct of  all  the  second  terms,  and  dividing  by  the  product  of  all 
the  first  terms.  Or,  which  amounts  to  the  same  thing,  that  vvc 
have  only  to  make  the  following  proportion  ;  as  the  product  of 
all  the  first  terms  is  to  the  product  of  all  the  second  terms,  so  is 
the  given  number  of  rubles  to  the  number  of  ducats  payable  at 
Berlin, 

428.  This  calculation  is  abridged  still  more,  when  amongst 
the  first  terms  some  are  found  that  have  common  divisors  with 
some  of  the  second  terms ;  for,  in  this  case,  we  destroy  those 
terms,  and  substitute  the  quotient  arising  from  the  division  by 
that  common  divisor.  The  preceding  example  will,  in  this 
manner,  assume  the  following  form.=^ 


les^. 

:     X%^^  siiY.  1000  rubles 

0. 

1  Dutch  flor. 

3. 

^  Dutch  dollars. 

100. 

142  dollars. 

3. 

:              1  ducat. 

IP3,  21. 

.    ;^,X;bj?i  ducats. 

6300 

2698  =  W0  :— 

7)  26980. 

9)    3854  (2 

428  (2.  MsweVf  428Jf  ducats* 
429.  The  method,  which  must  be  observed,  in  using  the  rule, 
of  reduction,  is  this ;  we  begin  with  the  kind  of  money  in  ques- 
tion, and  compare  it  with  another,  which  is  to  begin  the  next 
relation,  in  which  we  compare  this  second  kind  with  a  third, 
and  so  on.  Each  relation,  therefore,  begins  with  the  same  kind, 
as  the  preceding  relation  ended  with.  This  operation  is  con- 
tinued, till  we  arrive  at  the  kind  of  money  which  the  answer 
requires ;  and,  at  the  end,  we  reckon  the  fractional  remainders. 


•  Divide  the  1st  and  6th  by  2,  the  3d  and  12th  by  20,  the  5th  and  12th 
(which  is  now  5)  by  5,  also  the  2d  and  11th  by  5. 


Chap.  8.  Of  Compotuid  Quantities,  ISf 

430.  Other  examples  arc  added  to  facilitate  the  practice  of 
this  calculation. 

If  ducats  gain  at  Hamhurg  1  per  cent,  on  two  dollars  hanco  ; 
that  is  to  say,  if  50  ducats  are  worth,  not  100,  hut  101  dollars 
banco ;  and  if  the  exchange  between  Hamburgh  and  Konigs- 
berg  is  119  drachms  of  Poland  ;  that  is,  if  1  dollar  banco  gives 
119  Polish  drachms,  how  many  Polish  florins  will  1000  ducats 
give  ?  '^ 

30  Polish  drachms  make  1  Polish  florin. 
Ducat  1  :         ^  doll,  B°.  1000  due. 

/j0^,  50  :      101  doll.  B°. 
1         :     119  Pol.  dr. 
30         :  1  Pol.  flor. 


I5j^0         ;  12019=  lOj^fj^duc. 


5)  120190. 

5)  40063  (I. 

8012  (3.     Answer,  8012  I  p.  fl, 
431.  We  may  abridge  a  little  further,  by  writing  the  number, 
which  forms  the  third  term,  above  tlie  second  row  ;  for  tiien  the 
product  of  the  second  row,  divided  by  the  pmduct  of  the  first 
row,  will  give  the  answer  sought. 

Question,  Ducats  of  Amsterdam  are  brought  to  Lejpsick, 
having  in  the  former  city  the  value  of  5  flor.  4  stivers  current ; 
that  is  to  say,  1  ducat  is  worth  104  stivers,  and  5  ducats  are 
worth  26  Dutch  florins.  If,  therefore,  the  agio  of  the  bank*  at 
Amsterdam  is  5  per  cent,  that  is,  if  105  currency  arc  equal  to 
100  banco,  and  if  the  exchange  from  Leipsick  to  Amsterdam, 
in  bank  money,  is  1331  per  cent,  tliat  is,  if  for  100  dollars  wc 
pay  at  Leipsick  133^  dollars ;  lastly,  2  Dutch  dollars  making 
5  Dutch  florins ;  it  is  required  to  find  how  many  dollars  we 
must  pay  at  Leipsick,  according  to  these  exchanges,  for  1000 
ducats  ? 

*  The  difference  of  value  between  bank  money  and  current  money. 
18 


IBS  Algebra.  Sect.  3, 

/,  X000  ducats. 


Ducats      0 

26  fl.  Dutch  cuiT. 

4,  ^0,  i^i  flor.  Dutch  banco. 
:               533  doll,  of  Leipsick. 
:                   ^  doll,  banco. 

21 

:       3)  55432  (1. 

7)  18477  (4. 

2639. 
Jnswer,  2639^1  dollars,  or  2639  dollars  and  15  drachms. 


CHAPTER  IX. 

Of  Compound  Relations, 

432.  Compound  relations  are  obtained,  by  multiplying  the 
terms  of  two  or  more  relations,  the  antecedents  by  the  antece- 
dents, and  the  consequents  by  the  consequents  ;  we  say  then, 
that  the  relation  between  those  two  products  is  compounded  of 
the  relations  given. 

Thus,  the  relations  a  :  b,  c  :  d,  e  :  /,  give  the  compound  rela- 
tion ace  :  bdf.^ 

433.  A  relation  continuing  always  the  same,  when  we  divide 
both  its  terms  by  the  same  number,  in  order  to  abridge  it,  we 
may  greatly  facilitate  the  above  composition  by  comparing  the 
antecedents  and  the  consequents,  for  the  purpose  of  making 
such  reductions  as  we  performed  in  the  last  chapter. 

For  example,  we  find  the  compound  relation  of  the  following 
given  relations,  thus ; 

*  Each  of  these  three  ratios  is  said  to  be  one  of  the  roots  of  the  compound 
ratio. 


Chap.  9.  0/"  Compound  Quantities,  139 

Relations  given, 

12  :  25,    28  :  33,  and  55  :  56, 

/^,^2     :       5,    i^. 

^*  :    ^,    Ji^. 


2     :       5. 
So  that  2  :  5  is  the  compound  relation  required. 

434.  The  same  operation  is  to  be  performed,  when  it  is  re- 
quired to  calculate  generally  by  letters  ;  and  the  most  remark- 
able case  is  that,  in  which  each  antecedent  is  equal  to  the 
consequent  of  the  preceding  relation.    If  the  given  relations  are 

a  :  h 
h'.G 
c  :  d 
d  :  e 
e  :  a 
the  compound  relation  is  1  :  1. 

435.  The  utility  of  these  principles  will  be  perceived,  when 
it  is  observed,  that  the  relation  between  two  square  fields  is 
compounded  of  the  relations  of  the  lengths  and  the  breadths. 

Let  the  two  fields,  for  example,  be  A  and  B  ^  let  A  have  500 
feet  in  length  by  60  feet  in  breadth,  and  let  the  length  of  B  be 
360  feet,  and  its  breadth  100  feet;  the  relation  of  the  lengths 
will  be  500  :  360,  and  that  of  the  breadths  60  :  100.  So  that 
we  have 

i,5     :     6,^0^. 


5     :     6 
Wheixifore  the  field  A  is  to  the  field  B,  as  5  to  6. 

436.  ,^nother  examjjle.  Let  the  field  A  be  720  feet  long,  88 
feet  broad  ,•  and  let  the  field  B  be  660  feet  long,  and  90  feet 
broad ;  the  relations  will  be  compounded  in  the  following  man- 
ner: 

Relation  of  the  lengths,  /^JZ>,  8       :     15,i)/i,0f(ji 

Relation  of  the  breadths,  |^,  ^,  2     :  0 

Relation  of  the  fields  A  and  B,     16  :  15. 


140  •        Mgehra.  ^  Sect.  S. 

437'.  Further,  if  it  be  required  to  compare  two  chambers  with 
respect  to  the  space,  or  contents,  we  observe  that  that  I'clation 
is  compounded  of  three  relations  ;  namely,  of  tliat  of  the 
len.^ths,  that  of  the  breadths,  and  that  of  the  heights.  Let  there 
be,  for  example,  the  chamber  A,  whose  length  =  36  feet,  breadth 
=  16  feet,  and  height  =  14  feet,  and  the  chamber  B,  whose 
length  =  42  feet,  breadth  =  24  feet,  and  height  =  10  feet ;  we 
shall  have  these  three  relations : 

For  the  length    0,0         :    t,  0. 

For  the  breadth  /^,  /,  2     :    0,  ^^. 

For  the  height    U,  2         :     5,  /0. 


4         :  5 

So  that  the  contents  of  the  chamber  A  :  contents  of  the  cham- 
ber B,  as  4  :  5. 

438.  When  the  relations,  which  we  compound  in  this  manner, 
are  equal,  there  result  multiplicate  relations.  Namely,  two 
equal  relations  give  a  duplicate  ratio,  or  ratio  of  the  squares '; 
three  equal  relations  produce  the  triplicate  ratio,  or  ratio  of  the 
cubes f  and  so  on.  For  example,  the  relations  a  :  h  and  a  :  h  give 
the  compound  relation  aa  :  lib;  wherefore  we  say,  that  the 
squares  are  in  the  duplicate  ratio  of  their  roots.  And  the  ratio 
a  :  b  multiplied  thrice,  giving  the  ratio  a*  :  6^,  we  say  that  the 
cubes  are  in  the  triplicate  ratio  of  their  roots. 

439.  Geometry  teaches,  that  two  circular  spaces  are  in  the 
duplicate  relation  of  their  diameters;  this  means,  that  they  are 
to  each  other  as  the  squares  of  their  diameters; 

Let  A  be  a  circular  space,  having  the  diameter  =  45  feet,  and 
B  another  circular  space,  whose  diameter  =  SO  feet ;  the  first 
space  will  be  to  the  second  as  45  x  45  to  30  x  30 ;  or,  com- 
pounding these  two  equal  relations, 

//,J^,  3     :     2,0,0. 
//,^,  3     :     2,/,X;2f. 

9     :      4. 
Wherefore  the  two  areas  are  to  each  other  as  9  to  4. 

440.  It  is  also  demonstrated,  that  the  solid  contents  of  spheres 
are  in  the  ratio  of  the  cubes  of  the  diameters.    Thus,  the  diame- 


Chap.  9.  Of  Compmnd  Quantities,  141 

ter  of  a  globe  A,  being  1  foot,  aad  the  diameter  of  a  globe  B, 
being  £  feet,  the  solid  contents  of  A  will  be  to  those  of  B,  as 
13  :  £3  ;  or,  as  1  to  8. 

If,  therefore,  the  spheres  are  formed  of  the  same  substance, 
the  sphere  B  will  weigh  8  times  as  much  as  the  spiiere  A. 

441.  It  is  evident,  that  we  may,  in  this  manner,  find  the 
weight  of  cannon  balls,  their  diameters,  and  the  weight  of  one, 
being  given.  For  example,  let  there  be  the  ball  A,  whose 
diameter  =  £  inches,  and  wei.i^ht  =  5  pounds ;  and,  if  the 
weight  of  another  ball  be  i-equired,  whose  diameter  is  8  inches, 
we  have  this  proportion,  2^  :  S^  =  5  to  the  fourth  term,  S20 
pounds,  which  gives  the  weight  of  the  ball  B.  For  another  ball 
C,  whose  diameter  =  15  inches,  we  should  have, 

9.^  :  15^  =  5  : . . . .  Answer^  2109|  lb. 

CL  C 

442.  When  the  ratio  of  two  fractions,  as  -r  :  -15  is  requir- 
ed, w^e  may  always  express  it  in  integer  numbers  ;  for  we  have 
only  to  multiply  the  two  fractions  by  bd,  in  order  to  obtain  tlio 
ratio  axl  :  be,  which  is  equal  to  the  other ;  from  w^hich  results  the 

fl         c 

propoi-tion  -7  :  -^  =  ad  :  he.  If,  therefore,  ad  and  be  have  com- 
mon divisors,  the  ratio  may  be  reduced  to  less  terms.  Thus, 
It  •  If  =  15  X  3^  :  £4  X  £5  =  9  :  10. 

443.  If  we  wished  to  know  the  ratio  of  the  fractions  —  and  -7, 

a  0 

it  is  e^ ident,  that  we  should  have  —  :  —  =  b  :  a;  which  is  ex- 

(i      0 

pressed  by  saying,  that  two  fraetions,  whieh  have  unity  for  their 
numeratoTf  are  in  the  reciproeal,  or  inverse  ratio  of  their  denomi- 
nators.    The  same  may  be  said  of  two  fractions,  which  have  any 

c       c 
common  numerator  ;  for  —  :  ~  =zb  :  a.  But  if  two  fractions  have 

A 

their  denominators  equal,  as  —  :  — ,  they  are  in  tJie  direct  ratio  of 

the  numerators ;  namely,  as  a  :  b.  Thus,  | :  ^\  =1  ^^:  j\  z=  6:  S 
=  £  :  1,  and  V°  :  V  =  ^^  •  1^,  or,  =  2:3. 

444.  It  is  observed,  that  in  the  free  descent  of  bodies,  a  body 


lU  ^'       Mgebru.  Sect.  3. 

falls  16*  feet  in  a  second,  that  in  two  seconds  of  time  it  falls 
from  the  height  of  64  feet,  and  that  in  three  seconds  it  falls  144 
feet ;  hence  it  is  concluded,  that  the  heights  are  to  one  another 
as  the  squares  of  the  times ;  and  that,  reciprocally,  the  times 
are  in  the  sub  duplicate  ratio  of  the  heights,  or  as  the  square 
roots  of  the  heights. 

If,  therefore,  it  be  required  to  find  how  long  a  stone  must 
take  to  fall  from  the  height  of  2304  feet ;  we  have  16  :  2304  =  1 
to  tiie  square  of  the  time  sought.  So  that  the  square  of  the  time 
sought  is  144 ;  and,  consequently,  the  time  required  is  12  seconds. 

445.  It  is  required  to  find  how  far,  or  through  what  height, 
a  stone  will  pass,  by  descending  for  the  space  of  an  hour ;  that 
is,  3600  seconds.  We  say,  therefore,  as  the  squares  of  the  times, 
that  is,  I*  :  36002  ;  so  is  the  given  height  =16  feet,  to  the 
height  required. 

1  :  12960000  t=  16  : . . . .  207360000  height  required. 
16 


77760000 
1296 


207360000 
If  we  now  reckon  18000  feet  for  a  league,  we  shall  find  this 
height  to  be  10800  ;  and,  consequently,  nearly  four  times  greater 
than  tlie  diameter  of  the  earth. 

446.  It  is  the  same  with  regard  to  the  price  of  precious  stones, 
which  are  not  sold  in  the  proportion  of  their  weight ;  every 
body  knows  that  their  prices  follow  a  much  greater  ratio.  The 
rule  for  diamonds  is,  that  the  price  is  in  the  duplicate  ratio  of 
the  weight,  that  is  to  say,  the  ratio  of  the  prices  is  equal  to  the 
square  of  the  ratio  of  the  weights.  The  weight  of  diamonds  is 
expressed  in  carats,  and  a  carat  is  equivalent  to  4  grains ;  if, 
therefore,  a  diamond  of  one  carat  is  worth  10  livres,  a  diamond 
of  100  carats  will  be  worth  as  many  times  10  livres,  as  the 
square  of  100  contains  1  ;  so  that  we  shall  have,  according  to 
the  rule  of  three, 

*  15  is  used  in  the  orig-inal,  as  expressing-  the  descent  in  Paris  feet.  It  is 
here  altered  to  English  feet. 


ti^e 


Cliap.  9.  Of  Campmmd  ^anttnes,  14S 

12     :  100*  =  10  livrcs, 

or  1       :      10000    =  10  : . . . .  Answer,  100000  livrcs. 
There  is  a  diamond  in  Portugal,  which  weighs  1680  carats;  its 
price  will  be  found,  therefore,  hy  making 
12  ;        16802  =  10  liv:  ....or 
1     :  2822400   =  10  :  282M000  liv. 
447.  Th0  posts  or  mode  of  travelling  in  France,  furnish  exam- 
ples of  compound  ratios,  as  the  price  is  according  to  the  com- 
pound ratio  of  the  number  of  horses,  and  the  number  of  leagues, 
or  posts.     For  example,  one  horse  costing  20  sous  per  post,  it 
is  required  to  find  how  much  is  to  be  paid  for  28  horses  and  4| 
posts. 

We  write  first  the  ratio  of  the  horses,  1     :     28, 

Under  this  ratio  we  put  that  of  the  stages  or  posts,  2     :       9, 


And,  compounding  the  two  ratios,  we  have  2     :    252, 

Or,  1  :  126  =  1  livre  to  126  francs  or  42  crowns. 

Another  question.     If  I  pay  a  ducat  for  eight  horses,  for  3 

German  miles,  how  much  must  I  pay  for  thirty  horses  for  four 

miles  ?  The  calculation  is  as  follows : 
I,/     :      5,//,^^, 


1      :     5,  =  1  ducat  :  the  4th  term,  which  will  be 
5  ducats. 
448.  The  same  composition  occurs,  when  workmen  are  to  be 
paid,  since  those  payments  generally  follow  the  ratio  compound- 
ed of  the  number  of  workmen,  and  that  of  the  days  which  they 
have  been  employed. 

If,  for  example,  25  sous  per  day  be  given  to  one  mason,  and 
it  is  required  to  find  what  must  be  paid  to  24  masons  who  liave 
worked  for  50  days  ;   we  state  this  calculation  ; 
1  ;  24 
1  :  50 


1  :   1200  =  25  : 1500  francs, 

25 


20)  30000  (1506. 


144  ^     Algebra,  Sect.  3. 

As,  in  such  examples,  live  things  are  given,  the  rule,  which 
serves  to  resolve  them,  is  sometimes  called,  in  books  of  aritlr- 
metic  The  Rule  of  Five. 


CHAPTER  X. 

Of  Geometrical  Progessions, 

449.  A  SERIES  of  numbers,  which  are  always  becoming  a 
certain  number  of  times  greater,  or  less,  is  called  a  geometiical 
progressioiif  because  each  term  is  constantly  to  the  following  one  in 
the  same  geometrical  ratio.  And  the  number  which  expresses 
how  many  times  each  term  is  greater  than  the  preceding,  is 
called  the  exponent.  Thus,  when  the  first  term  is  1  and  the 
exponent  =  2,  the  geometrical  progression  becomes. 

Terms       123456789       &:c. 

Prog,        1,     2,   4,    8,  16,    32,    64,   128,  256,    &c. 
The  numbers  1,  2,  3,  &c.  always  marking  the  place  which  each 
term  holds  in  the  progression. 

450.  If  we  suppose,  in  general,  the  first  term  =  a,  and  the 
exponent  =  b,  we  have  the  following  geometrical  progression  ; 

1,   2,      3,      4,       5,       6,       7,       8    'n 

Prog,  fl,  abi  ab^,  ah^,  ab^^  ab^,  ab^,  ah''  ....  ai"~^ 
So  that,  when  this  progression  consists  of  n  terms,  the  last 
term  is  =  a6"~^  We  must  remark  here,  that  if  the  exponent  & 
be  greater  than  unity,  the  terms  increase  continually ;  if  the  ex- 
ponent &  =  1,  the  terms  are  all  equal ;  lastly,  if  the  exponent  b 
be  less  than  1,  or  a  fraction,  the  terms  continually  decrease. 
Thus,  when  a  =  1  and  6  =  ^»  we  have  this  geometrical  progres- 
sion ] 

1     1X11111        i.p 

451.  Here  therefore  we  have  to  consider ; 

I.  The  first  term,  which  wo  have  called  a, 
II.  The  exponent,  which  wc  call  b. 

III.  The  number  of  terms,  which  we  have  expressed  by  n, 

IV.  The  last  term,  which  we  have  found  =  at"~^ 

So  that,  when  the  three  first  of  these  are  given,  the  last  term  is 


Chap.  10.  Of  Compound  quani/^s.  145 

found,  by  multiplying  the  n  -  1  power  of  b,  or  6^-^  by  tbe  first 
term  a.  ,. 

If,  therefore,  the  50tli  term  of  the  geometrical  progression  1, 

2,  4,  8,  &c.  were  required,  we  should  have  a  =  1,  6  :=  2,  and  n 
=  50  ;  consequently  the  50th  term  =  2*  \  Now  2»  being  ^  512  ; 
210  will  ije  _  1024.  Wherefore  the  square  of  2*%  or  2«o,  = 
1048576,  and  the  square  of  this  number,  or  1099511627776  = 
2*0.  Multiplying  therefore  this  value  of  2*°  by  2^,  or  by  512, 
we  have  2*»  equal  to  562949953421312. 

452.  One  of  the  principal  questions,  which  occurs  on  this 
subject,  is  to  find  the  sum  of  all  the  terms  of  a  geometrical  pro- 
gression ;  we  shall  therefore  explain  the  method  of  doing  this. 
Let  there  be  given,  first,  the  following  progression,  consisting  of 
ten  terms ; 

1,  2,  4,  8,  16,  32,  64,  128,  256,  512. 
the  sum  of  which  we  shall  represent  by  s,  so  that,  s  =  1  -|-  2  + 
4  4-  8  +  16  +  32  -f-  64  +  128  +  256  +  512  ;  doubling  both  sides, 
we  shall  have  2  s  =  2  -f-  4  -f  8  +  l6  +  32  +  64  +  128  +  256  + 
512  -f  1024.  Subtracting  from  this  the  progression  represented 
by  s,  there  remains  s  =  1024  —  1  =  1023  ;  wherefore  the  sum 
required  is  1023. 

453.  Suppose  now,  in  the  same  progression,  that  the  number 
of  terms  is  undetermined  and  =  n,  so  that  the  sum  in  question, 
or  s,  =  1  -f  2  -f  2*  -f  2^  4-  2*  . . . .  2"~^  If  we  multiply  by  2, 
we  have  28  =  2+2*  +2^  +2*  -...y,  and  subtracting  from 
this  equation  the  preceding  one,  we  have  s  =  2n —  l.  We  see, 
therefore,  that  tlie  sum  required  is  found,  by  multiplying  the  last 
term,  2"-^  by  the  exponent  2,  in  order  to  have  2",  and  subtract- 
ing unity  from  that  product. 

454.  This  is  made  still  more  evident  by  the  following  exam- 
ples, in  which  we  substitute  successively,  for  n,  the  numbers  1,2, 

3,  4,  &c. 

1=1;  1+2=3;   1+2+4  =  7;  1+2+4  +  8=15; 
1+2+4  +  8  +  16  =  31;   1  +  2 +4  +  8  +  16  +  32  =  63,  &c. 

455.  On  this  subject  the  following  question  is  generally  pro- 
posed. A  man  offers  to  sell  his  horse  by  the  nails  in  his  shoes, 
which  are  in  number  32  ;  he  demands  1  liard  for  the  first  nail, 

19 


146  .^   Mgebra, 


Sect.  S. 


2  for  the  second,  4  for  the  third,  8  for  the  fourth,  and  so  on,  de- 
manding ^or  each  nail  twice  the  price  of  tlie  preceding.  It  is 
required  to  find  what  would  be  the  price  of  the  horse  ? 

This  question  is  evidently  reduced  to  finding  the  sum  of  all 
the  terms  of  the  geometrical  progression  1,  2,  4,  8,  16,  &c,  con- 
tinued to  the  32d  term.  Now,  this  last  term  is  2^^  ^  ;  .and,  as  we 
have  already  found  9.^^  =  1048576,  and  2^°  =  1024,  we  shall 
have  2»o  X  210  _  230  equal  to  1073741824  ;  and  multiplying 
again  by  2,  the  last  term  2^*  =  2147483648;  doubling  there- 
fore this  number,  and  subtracting  unity  from  the  product,  the 
sum  required  becomes  4294967295  liards.  These  liards  make 
10737418231  sous,  and  dividing  by  20,  we  have  53687091  livrcs, 

3  sous,  9  deniers  for  the  sum  requii-ed. 

456.  Let  the  exponent  now  be  =  3,  and  let  it  be  required  to 
find  the  sum  of  the  geometrical  progression  1,  3,  9,  27,  81,  243, 
729,  consisting  of  7  terms.     Suppose  it  =  s,  so  that, 

.     s  =  1  -I-  3  +  9  +  27  +  81  +  243  -I-  729. 
we  shall  then  have,  multiplying  by  3, 

3s  =  3  +  9  +  27  -f-  81  +  243  +  729  +  2187. 
and  subtracting  the  preceding  series,  we  have  2s  =  2187  —  1  = 
2186.     So  that  the  double  of  the  sum  is  2186,  and  consequently 
the  sum  required  =  1093. 

457.  In  the  same  progression,  let  the  number  of  terms  =  n,  and 

the  sum  =  s  ;  so  that  s  =  1  +3  +  3»  +  3^  +  3^  + 3"-i. 

If  we  multiply  by  3,  we  have  3s  =  3  +  3*  +  3*  -|-  3*  + 3«. 

Subtracting  from  this  the  value  of  s,  as  all  the  terms  of  it, 
except  the  first,  destroy  all  the  terms  of  the  value  of  3s,  except 

3"  —  1 
the  last,  we  shall  have  2s  =  3"  —  1 ;  therefore  s  =   — - —    So 

that  the  sum  required  is  found  by  multiplying  the  last  term  by 
3,  subtracting  1  from  the  product,  and  dividing  the  remainder 
by  2.    This  will  appear,  also,  from  the  following  examples; 

1  +  3  +  9  +  27  =  ^^^^~^  =  40  ;    1  +  3   +  9   +  27  +  81  = 


Chap.  10.  Of  Compmnd  Quantities,  14f 

458.  Let  us  now  suppose,  generally,  the  first  term  =  a,  the 
exponent  =  5,  the  number  of  terms  =  n,  and  their  sum  =  s,  sa 
that, 

s  =  a  -f-  aft  4-  a&3  +  aft^  +  aft*  +  . . . .  aZ>«"\ 

If  we  multiply  by  ft,  we  have 

6s  =  fl6  -f  a6*  +  aft*  +  oft*  +  oft*  +  . . . .  a6",  and  subtract- 
ing the  above  equation,  there  remains  (6  —  1)  s  =  a6»  —  fl , 

whence  we  easily  deduce  the  sum  required  s  =  — — — .    Catise- 

qitently,  the  sum  of  any  geometrical  progression  is  founds  by  miUti- 
plying  tlie  last  term  by  the  exponent  of  the  progression,  subtracting 
the  first  term  from  the  jn^oduct,  and  dividing  the  remainder  by  the 
exponent  minus  unity, 

459.  Let  thereby  a  geometrical  progression  of  seven  terms, 
of  which  the  hrst  =  3  ;  and  let  the  exponent  be  =  2  ;  we  shall 
then  have  a  =  S,  6  =  2,  and  n  =  7  ;  wherefore  the  last  term  = 
3  X  2*,  or  3  X  64  =  192 ;  and  the  whole  progression  will  be 

3,  6,  12,  24,  48,  96,   192. 
Further,  if  we  multiply  the  last  term  192  by  the  exponent  2, 
we  have  384;  subtracting  the  first  term,  there  remains  381; 
and  dividing  this  by  6  —  1,  or  by  1,  we  have  381  for  the  sum  of 
the  whole  progression. 

460.  Again,  let  there  be  a  geometrical  progression  of  six 
terms  ;  let  4  be  the  first,  and  let  the  exponent  be  =  |.  The 
progression  is 

4     f;     Q     27       81      243 

If  we  multiply  this  last  term  ^4^  by  the  exponent  |,  we  shall 
have  \Y  J  t^6  subtraction  of  the  first  term  4  leaves  the  remain- 
der \Y>  which,  divided  by  6  —  1  =  i,  gives  «|*  =  83|. 

461.  When  the  exponent  is  less  than  1,  and  consequently, 
when  the  terms  of  the  progression  continually  diminish,  the  sum 
of  such  a  decreasing  progression,  which  would  go  on  to  infinity, 
may  be  accurately  expressed. 

For  example,  let  the  first  term  =1,  the  exponent  =  |,  and 
the  sum  =  s,  so  that 

«  =  1  +  I  +  i  +i  +  tV  +  A  +  tV  +  &c. 
ad  infinitum. 


148  "^^rr^iAlgebra.  Sect.  3. 

If  we  multiply  by  2,  we  have 

ad  infinitum. 

And,  subti'actinj^  the  preceding  progression,  there  remains 
s  =  2  for  the  sum  of  the  proposed  infinite  progression, 

4G2.  If  tlie  first  term  =  1,  the  exponent  =  -|,  and  the  sum 
=  s  i  so  that 

5=1  +  I  4-  1  -I-  -1^  4-  _i_  -f.  &c.  ad  infinitum. 

Multiplying  the  whole  by  ^,  we  have 

Ss  =  3  -f  1  4-  1  +  1  +  ^1^  -I-  &c.  ad  infinitum ; 
and  subtracting  the  value  of  s,  there  remains  2s  =  S;  wherefore 
the  sum  s  =  i^. 

463.  Let  there  be  a  progression,  whose  sum  =  s,  first  term 
=  2,  and  exponent  =  | ;  so  that  s  =  2  +  |  + 1  +  |J  +  ^^g.  + 
&c.  ad  infinitum. 

Multiplying  by  |,  we  have  |  s  =  |  +  2  +  |  +  |  +  ||  +  ^V^ 
+  &c.  ad  infinitum.  Subtracting  now  the  progression  s,  there 
remains  ^  s  =  | ;  wherefore  the  sum  required  =  8. 

464.  If  we  suppose,  in  general,  the  first  term  =  a,  and  the 

exponent  of  the  progression  =  — ,  so  that  this  fraction  may  be 

less  than  1,  and  consequently  c  greater  than  b  ;  the  sum  of  the 
progression  carried  on,  ad  infinitum,  will  be  found  thus  5 

Make  s  z=  a  -4 -4 —  4-  dec. 

c         cc  c^  c^     ' 

Multiplying  by  — ,  we  shall  have 

b         ab   .    ab^        ab^        ab"^        c         j  •  /»    w 

—  s  = — —  H — —  -4 r-  4-  6cc.  ad  infinitum. 

c  c  ^  c^    ^  c^    ^    c'*'    ^ 

And,  subtracting  this  equation  from  the  preceding,  there  re- 
mains CI )  s  =  a. 

a 
Consequently  s  =  1  —  b, 
c 
If  we  multiply  both  terms  of  this  fraction  by  c,  we  have 

s  =  — .    The  sum  of  the  infinite  geometrical  progression 


Chap.  10.  Of  Compmvnd  ^antities,  149 

proposed  is,  therefore,  found  by  dividing  the  first  term  a  by  1 
minus  the  exponent,  or  by  multiplying  the  first  term  a  by  the 
denominator  of  the  exponent,  and  dividing  the  product  by  the 
same  denominator  diminished  by  the  numerator  of  the  exponent. 
465  In  the  same  manner,  we  find  the  sums  of  progressions, 
the  terms  of  which  are  alternately  affected  by  the  signs  -f-  and 
— .     Let,  for  example, 

ab    .     ab^         ab^         ah*  „ 

Multiplying  by  — ,  we  have 

b     _ab         ab^         ab^  ab*  ^ 

And,  adding  this  equation  to  the  preceding,  we  obtain  (1  ^-  —) 

s  =  a.    Whence  we  deduce  the  sum  required  s  = -,  or  ^  = 

1  4-  & 

ac  c 


c  -f  6' 

466.  We  see,  then,  that  if  the  first  term  a  =  |,  and  the  expo- 
nent  =  f ,  that  is  to  say,  6  =  2  and  cz=:  5,  we  shall  find  the  sum 
of  the  progression  34./-^+  ^\%  +  ^%\  +  &c.  =  1 ;  since,  by 
subtracting  the  exponent  from  1,  there  remains  |,  and  by  divid- 
ing the  first  term  by  that  remainder,  the  quotient  is  1. 

Further,  it  is  evident,  if  the  terms  be  alternately  positive  and 
negative,  and  the  progression  assume  this  form  ; 

3    6         1        12      24        iJ^p 

T  IlJ  ^  T2T  6  2T  4-  **^* 

the  sum  will  be 


1+11        7 
c 
467.  »Brwther  example.     Let  there  be  proposed  the  infinite 
progression. 

The  first  term  is  here  j^,  and  the  exponent  is  j\.  Subtract- 
ing this  last  from  1,  there  remains  j%,  and,  if  we  divide  the 
first  term  by  this  fraction,  we  have  ^  for  the  sum  of  the  given 
progi'ession.  So  that  taking  only  one  term  of  the  progression, 
namely,  j%,  the  error  would  be  ^V» 


150  -Mgebra.  Sect.  3. 

Taking  two  terms,  ^^^  +  ^'^  ^  ^,j_^^  ^^levQ  would  stUl  be 
wanting  ^^^  to  make  the  sum  =  -l. 

468.  Mother  example.  Let  there  be  given  the  infinite  pro- 
gression, 

^  "^  TTT  "f-  T¥o   +  t/iTTT  +  To7"o  0"  +  &C. 

The  first  term  is  9,  the  exponent  is  ^V*  So  that  1,  minus  the 
exponent,  =  -»^  ,•  and  _l  =  10,  the  sum  required. 

This  series  is  expressed  by  a  decimal  fraction,  thus  9,9999999, 
&c. 


CHAPTER  XL 

Of  Infinite  Decimal  Fractions. 

469.  It  will  be  very  necessary  to  shew  how  a  vulgar  fraction 
may  be  transformed  into  a  decimal  fraction ;  and,  conversely, 
how  we  may  express  the  value  of  a  decimal  fraction  by  a  vulgar 
fraction. 

470.  Let  it  be  required^  in  general,  to  change  the  fraction  --,  into 

a  decimal  fraction ;  as  this  fraction  expresses  the  quotient  of  the 
division  of  the  numerator  a  by  the  denominator  b,  let  us  write, 
instead  of  a,  tJie  quantity  a,0000000,  whose  value  does  not  at  all 
differ  from  that  of  a,  since  it  contains  neither  tenth  parts,  nor  huu' 
dredth  parts,  S^c,  Let  us  now  divide  this  quantity  by  the  number  b, 
according  to  the  common  rules  of  division,  observing  to  put  the  point 
in  the  proper  place,  which  separates  the  decimal  and  the  integers. 
This  is  the  whole  operation,  which  we  shall  illustrate  by  some 
examples. 

Let  there  be  given  first  the  fraction  ^,  the  division  in  deci- 
mals, ,will  assume  this  form, 

2)  1,0000000  _  J. 
0,5000000  ~"  "2* 

Hence  it  appears,  that  ^  is  equal  to  0,5000000  or  to  0,5 ; 
which  is  sufiicicntly  evident,  since  this  decimal  fraction  repre- 
sents j\,  which  is  equivalent  to  ^-. 


Chap.  11.  Of  Compound  ^antities,  151 

471.  liCt  -I  be  the  given  fraction,  and  we  have, 

3)  1,0000000    „        _  _1 
0,3333333         *  3* 

This  shews,  that  the  decimal  fraction,  whose  value  is  =  ^, 
cannot,  strictly,  ever  be  discontinued,  and  that  it  goes  on  ad 
infinitum,  repeating  always  the  number  3.  And,  for  this  reason, 
it  has  been  already  shewn,  that  the  fractions  -^^  -f-  -|^  -f.  ^/^^ 
^  -j^477  ^^'  ^^  infinitum,  added  together,  make  4. 

The  decimal  fraction  which  expresses  the  value  of  |,  is  also 
continued  ad  infinitum  ;  for  we  have, 

3)  2.0000000   c       _  3 
0,b6(5t)b66     ^'  ""  ~3* 

And  besides,  this  is  evident  from  what  we  have  just  said, 
because  |  is  the  double  of  |. 

472.  If  1  be  the  fraction  proposed,  we  have 

4)  1,0000000   e      __2 

o^uoooo     • ""  T 
So  that  1  is  equal  to  0,2500000,  or  to  0,25  ;  and  this  is  evi- 
dent, since  ^\  -f-  -j  J^  =  ^Vtt  =  i- 
In  like  manner,  we  shoidd  have  for  the  fraction  |, 
4)  3,0000000  _   3 
0,7o00000  ~  "i* 
So  that  I  =  0,75  ;  and  in  fact  ^\  +  ^|^  =  ^\%  =  |. 
The  fraction  f  is  changed  into  a  decimal  fraction,  by  making 
4)  5,0000000  _  5 
1,2500000  """i* 

Now  1  +  ^VV  =  I- 

473.  In  the  same  manner,  ^  will  be  found  equal  to  0,2 ;  |  = 
=  0,4  ;  I  =  0,6 ;  I  =  0,8 ;  I  =  1  ;  I  =  1,2,  &c. 

When  the  denominator  is  6,  we  find  ^  =  0,1666666,  &c.  which 
is  equal  to  0,666666  —  0,5.  Now  0,666666  =  |  and  0,5  =  |, 
wherefore  0,1666666  =  |  —  |  =  ^. 

We  find,  also,  |  =  0,333333,  &c.  =  ^  ;  but  |  becomes 
0,5000000  =  i.  Further,  f  =  0,833333  ;=  0,333333  -f  0,5, 
that  is  to  say,  |  -f-  i  =  a. 

474.  When  the  denominator  is  7,  the  decimal  fractions  be- 
come more  complicated.  For  example,  we  find  |  =  0,142857, 
however  it  must  be  observed,  that  these  six  figures  are  lepeated 


152  ^Igehra.  Sect,  3. 

continually.  To  be  convinced,  therefore,  that  this  decimal 
fraction  precisely  expresses  tbe  value  of  ^,  we  may  transform  it 
it  into  a  geometrical  progression,  whose  first  term  is  =  y^-^^V/tto  » 
and  the  exponent  =  t<^^^(5^o77  ?    ^"^'   consequently,  the  sum 

1 42857 

(art.  464)  =      '^"""^^     (multiplying  both  terms  by  1000000) 

142^8  5  7    1 

—    9'5"9'5'9  9   T* 

475.  We  may  prove,  in  a  manner  still  more  easy,  that  the 
decimal  fraction  which  we  have  found  is  exactly  =  ^  ;  for,  sub- 
stituting for  its  value  the  letter  5,  we  have 

S  =  0,142857142857142857,  &C. 

10  S  =  1,  42857142857142857,  &c. 

100  S  =  14,  2857142857142857,  &c. 

1000  s  =  142,  857142857142857,  &C. 

10000  S  =  1428,  57142857142857,  &c. 

100000  s  =  14285,  7142857142857,  &c. 

1000000  s  =  142857,   142857142857,  &c. 

Subtract  s=  0,  142857142857,  &c. 


999999  s  =  142857. 

And,  dividing  by  999999,  we  have  s  =  ||||f  J  =  4-.    Where- 
fore the  decimal  fraction,  which  was  made  =  s,  is  =  ^. 

476.  In  the  same  manner  f  may  be  transformed  into  a  deci- 
mal fraction,  which  will  be  0,28571428,  &c.  and  this  enables  us 
to  find  more  easily  the  value  of  the  decimal  fraction  which  we 
have  supposed  =  s ;  because  0,28571428  &c.  must  be  the  double 
of  it,  and,  consequently,  =  2s.  For  we  have  seen  that 
100  S  —  14,28571428571  &c. 

So  that  subtracting      2  s  =    0,28571428571  &c. 


there  remains      98  s  =14 

wherefore         s  =  \-^  =  ^, 
We  also  find  4  =  0,42857142857  &c.  which,  according  to  our 
supposition,  must  be  =  3s  ;  now  we  have  found  that 
10  S  =  1,42857142857  &C. 
So  that  subtracting  3  s=  0,42857142857  &c. 


we  have  7  s  =  1,  wherefore  s  =  -4. 


Chap.  11.  Of  Compound  Quantities.  153 

477.  When  a  proposed  fraction,  therefore,  has  the  denomina- 
tor 7,  the  decimal  fraction  is  infinite,  and  6  figures  are  con- 
tinually repeated.  The  reason  is,  as  it  is  easy  to  perceive,  that 
when  we  continue  the  division  we  must  return,  sooner  or  hiter, 
to  a  remainder  which  we  have  had  already.  Now,  in  this  divi- 
sion, 6  different  numbers  only  can  form  the  remainder,  namely, 
1,  2,  3,  4,  5,  6  ;  so  that,  after  the  sixth  division,  at  furthest,  the 
same  figures  must  return  ;  but  when  the  denominator  is  such  as 
to  lead  to  a  division  without  remainder,  these  cases  do  not 
happen. 

478.  Suppose,  now,  that  8  is  the  denominator  of  the  fraction 
proposed  ;  we  shall  find  the  following  decimal  fractions  ; 

-I  =  0,125  ;  I  =  0,25  ;  |  =  0,375  ;  ^  =  0,5  ;  |  =  0,625  ;  |  = 
0,75  ;  I  =  0,875  ;  &c. 

If  the  denominator  be  9,  we  have  |  =0,111  &c.  |  =  0,222 
6cc.  I  =  0,333  kc. 

If  the  denominator  be  10,  we  have  j\  =  0,1  ;  ^^  =  0,2  ;  -^^  = 
0,3.  This  is  evident  from  the  nature  of  tlie  thing,  as  also  that 
^J^  =  0,01 ;  that  ^Vo  =  0^37  ;  that  ^VoV  =  0,256  ;  that  ^^%\^ 
=;  0,0024  &C. 

479.  If  11  be  the  denominator  of  the  given  fraction,  we  shall 
have  ri\  =  0,0909090  &c.  Now,  suppose  it  were  required  to 
find  the  value  of  this  decimal  fraction  ;  let  us  call  it  s,  we  shall 
have  sz=  0,090909,  and  10  s  =  00,909090;  further,  100  s  = 
9,09090.  If,  therefore,  we  subtract  from  the  last  the  value  of  s, 
we  shall  have  99  s  =  9,  and  consequently  s  =  -§?g  =  -^l^.  We 
shall  have,  also,  ^\  =  0,181818  &c.  -j\  =  0,272727  kc.  j\  = 
0,545454  &C. 

480.  There  is  a  great  number  of  decimal  fractions,  therefore, 
in  which  one,  two,  or  more  figures  constantly  recur,  and  which 
continue  thus  to  infinity.  Such  fractions  are  curious,  and  we 
shall  shew  how  their  values  may  be  easily  found. 

Let  us  first  suppose,  that  a  single  figure  is  constantly  repeat- 
ed, and  let  us  represent  it  hy  a,  so  that  s  =  Ofaaaaaaa,  We  have 
10  s  =  a^aaciaaaa 
and  subtracting      s  =  Q,aaaaaaa 


we  have     9  s  =  a ;  wherefore  s  =  —. 
20 


154  Jlgehra,  Sect.  3. 

When  two  iigures  are  repeated,  as  ah,  we  have  s  =  0,abababa, 
Therefore  100  s  =  ab^ababab ;  and  if  we  subtract  s  from  it,  there 

/ih 

remains  99s  =  ab ;  consequently  s  =  — . 

When  three  figures,  as  abc,  are  found  repeated,  we  have  s  = 
0,abcabcabc  ;    consequently,  1000  s  =  abc^abcabc  ;  and  subtract 

s  from  it,  there  remains  999  s  =  abc ;  wherefore  s  =  ■-— ,  and 

yyy 

so  on. 

Whenever,  therefore,  a  decimal  fraction  of  this  kind  occurs, 
it  is  easy  to  find  its  value.  Let  there  be  given,  for  example, 
0,296296,  its  value  will  be  |||  =  -^j,  dividing  both  terms  by  27* 

This  fraction  ought  to  give  again  the  decimal  fraction  pro- 
posed ;  and  we  may  easily  be  convinced  that  this  is  the  real 
result,  by  dividing  8  by  9,  and  then  that  quotient  by  3,  because 
27  =  3  X  9.     We  have 

9^  8,0000000 


3)  0,8888888 

0,2962962,  &c. 
which  is  the  decimal  fraction  that  was  proposed. 

481.  We  shall  give  a  curious  example,  by  changing  the  frac- 
tion  — ; — - — - — - — - — - — 7, ~,  into  a  decimal  frac- 

1  X2x3x4x5x6x7x8x9xl0 

tion.    The  operation  is  as  follows. 

2)  1,00000000000000 


3)  0,50000000000000 

4)  0,16666666666666 

5)  0,04166666666666 

6)  0,00833333333333 

7)  0,00138888888888 


Chap.  11.  Of  Cmnprnuul  Quantities.  155 

8)  0,00019841269841 


9)  0,00002480158730 
10)  0,00000275573192 


0,00000027557319. 


i 

-'^V- 


SEITlOiY  FOUllTH. 

OF  ALGEBRAIC  EQUATIONS,  AND  OF  THE  RESOLUTION  OF  THOSE  EQUATIONS. 

CHAPTER  I. 

Of  the  Solution  of  Problems  in  general, 

ARTICLE  482. 

The  principal  object  of  Algebra,  as  well  as  of  all  the  parts 
*of  Mathematics,  is  to  determine  the  value  of  quantities  which 
were  before  unknown.  This  is  obtained  by  considering  atten- 
tively the  conditions  given,  which  are  always  expressed  in 
known  numbers.  For  this  reason  Algebra  has  been  defined. 
The  science  which  teaches  how  to  determine  unknown  quantities  by 
means  of  known  quantities, 

483.  The  definition  which  we  have  now  given  agrees  with  all 
that  has  been  hitherto  laid  down.  We  have  always  seen  the 
knowledge  of  certain  quantities  lead  to  that  of  other  quantities, 
which  before  might  have  been  considered  as  unknown. 

Of  this,  addition  will  readily  furnish  an  example.  To  find 
the  sum  of  two,  or  more  given  numbers,  we  had  to  seek  for  an 
i^iknown  number  which  should  be  equal  to  those  known  num- 
bers taken  together. 

In  subtraction,  we  sought  for  a  number  which  should  be  equal 
to  the  difference  of  two  known  numbers. 

A  multitude  of  other  examples  are  presented  by  multiplica- 
tion, division,  tlie  involution  of  powers,  and  the  extraction  of 
roots.  The  question  is  always  reduced  to  finding,  by  means 
of  known  quantities,  another  quantity  till  then  unknown. 

484.  In  the  last  section  also,  different  questions  were  resolved, 
in  which  it  was  required  to  determine  a  number,  that  could  not 


Chap.  1.  Of  Compound  Quantities,  157 

be  deduced  from  the  knowledge  of  other  given  numbers,  except 
imder  certain  conditions. 

All  those  questions  were  reduced  to  finding,  by  the  aid  of 
some  given  numbers,  a  new  number  which  should  have  a  certain 
connexion  with  them ;  and  this  connexion  was  determined  by 
certain  conditions,  or  properties,  which  were  to  agree  with  the 
quantity  sought. 

485.  Wheji  we  have  a  question  to  resolve,  we  represent  the  num- 
ber sought  by  mie  of  the  last  letters  of  the  alphabet,  and  then  consi- 
der in  what  manner  the  given  conditions  can  form  an  equality 
between  two  quantities.  This  equality,  wliich  is  represented  by 
a  kind  of  formula,  called  an  equation,  enables  us  at  last  to  deter- 
mine the  value  of  the  number  sought,  and  consequently  to 
resolve  the  question.  Sometimes,  several  numbers  aie  sought ; 
but  they  are  found  in  the  same  manner  by  equations. 

486.  Let  us  endeavour  to  explain  this  further  by  an  example. 
Suppose  the  following  question,  or  problem  was  proposed. 

Twenty  persons,  men  and  women,  dine  at  a  tavern  ;  the  share 
of  the  reckoning  for  one  man  is  8  sous,*  tliat  for  one  woman  is 
r  sous,  and  the  whole  reckoning  amounts  to  7  livres  5  sous ; 
required,  the  number  of  men,  and  also  of  women  ? 

In  order  to  resolve  this  question,  let  us  suppose  that  the  num- 
ber of  men  is  =  ^ ;  and  now  considering  this  number  as  known, 
we  shall  proceed  in  the  same  manner  as  if  we  wished  to  try 
whether  it  corresponded  with  the  conditions  of  the  question. 
Now,  the  number  of  men  being  =  x,  and  the  men  and  women 
making  together  twenty  persons,  it  is  easy  to  determine  the 
number  of  the  women,  having  only  to  subtract  tliat  of  the  men 
from  20,  that  is  to  say,  the  number  of  women  =  20  —  x,  # 

But  each  man  spends  8  sous  ;  wherefore  x  men  spend  8,x  sous. 

And,  since  each  woman  spends  7  sous,  20  —  x  women  must 
spend  140  —  7x  sous. 

So  that  adding  together  ^x  and  140  —  7x,  we  see  that  the 
whole  20  persons  must  spend  140  -f  x  sous.  Now,  we  know 
already  how  much  they  have  spent ;  namely,  7  livres  5  sous ; 
or  145  sous ;  there  must  be  an  equality  therefore  between  14Q 

^  A  sous  is  ^  of  a  livre ;  a  livre  ^  of  a  crown  or  17  cents  6  mills. 


158  Jilgebi-a,  Sect  4. 

-f  X  and  145 ;  that  is  to  say,  we  have  the  equation  140  +  a;  = 
145,  and  thence  we  easily  deduce  a?  =  5. 
So  that  the  company  consisted  of  5  men  and  15  women. 

487.  Jlnother  question  of  the  same  kind. 

Twenty  persons,  men  and  women,  go  to  a  tavern  ;  the  men 
impend  24  florins,  and  the  women  as  much  ;  but  it  is  found  that 
each  man  has  spent  1  florin  more  than  each  woman.  Required, 
the  number  of  men  and  the  number  of  women  ? 

Let  the  number  of  men  =  x. 

That  of  the  women  will  be  =  20  —  x, 

Now  these  x  men  having  spent  24  florins,  the  share  of  each 

24 
man  is  —  florins. 

Further,  the  20  —  x  women  having  also  spent  24  florins,  the 

"24 

share  of  each  woman  is  — florins. 

20—^ 

But  we  know  that  the  share  of  each  woman  is  one  florin  less 

than  that  of  each  man ;  if,  therefore,  we  subtract  1  from  the 

share  of  a  man,  we  must  obtain  that  of  a  woman ;  and  conse- 

24                      24 
quently 1  =  — .  This,  therefore,  is  the  equation  from 

which  we  are  to  deduce  the  value  of  x.  This  value  is  not  found 
with  the  same  ease  as  in  the  preceding  question ;  but  we  shall 
soon  see  that  a;  =  8,  which  value  corresponds  to  the  equation ; 
for  \*  —  1  =  41  includes  the  equality  2  =  2. 

488.  It  is  evident  how  essential  it  is,  in  all  problems,  to  con- 
sider the  circumstances  of  the  question  attentively,  in  order  to 
deduce  from  it  an  equation,  that  shall  express  by  letters  the 
numbers  sought  or  unknown.  After  that,  the  whole  art  consists 
in  resolving  those  equations,  or  deriving  from  them  the  values 
of  the  unknown  numbers  5  and  this  shall  be  the  subject  of  the 
present  section. 

489.  We  must  remark,  in  the  first  place,  the  diversity  which 
subsists  among  the  questions  themselves.  In  some,  we  seek 
only  for  one  unknown  quantity  ;  in  others,  we  have  to  find  two, 
or  more  ;  and  it  is  to  be  observed,  with  regard  to  this  last  case, 
that  in  order  to  determine  them  all,  we  must  deduce  from  the 
circumstances,  or  the  conditions  of  the  problem,  as  many  equa- 
tions as  there  arc  unknown  quantities. 


Chap.  2.  Of  Comjmind  Quantities.  159 

490.  It  must  have  already  been  perceived,  that  an  equation 
consists  of  two  parts  separated  by  the  sign  of  equality^  =,  to 
shew  that  those  two  quantities  are  equal  to  one  another.  We 
are  often  obliged  to  perform  a  great  number  of  transformations  on 
those  tw  o  parts,  in  order  to  deduce  from  them  the  value  of  the 
unknown  quantity ;  but  these  transformations  must  be  all  found- 
ed on  the  following  principles  ;  that  two  quantities  remain  eqiud, 
whether  we  add  to  them,  or  subtract  from  them  equal  quanti- 
ties ;  whether  we  multiply  them,  or  divide  them  by  the  same  num- 
ber ;  whether  we  raise  them  both  to  the  same  power,  or  extract 
tlieir  roots  of  the  same  degree. 

491.  The  equations  which  are  resolved  most  easily,  are  those 
in  which  the  unknown  quantity  does  not  exceed  the  first  power, 
after  the  terms  of  the  equation  having  been  properly  arranged  : 
and  we  call  them  simple  equations,  or  equations  of  the  first  degree. 
But  if,  after  having  reduced  and  ordered  an  equation,  we  find  in 
it  the  square,  or  the  second  power  of  the  unknown  quantity,  it 
may  be  called  an  equation  of  the  second  degree,  which  is  more 
difficult  to  resolve. 


CHAPTER  II. 

Of  the  Resolution  of  Simple  Equations,  or  Equations  of  the  first 

degree, 

492.  When  the  number  sought,  or  the  unknown  quantity,  is 
represented  by  the  letter  x,  and  the  equation  we  have  obtained 
is  such,  that  one  side  contains  only  that  x,  and  the  other  simply 
a  known  number,  as  for  example,  x  =  25,  the  value  of  x  is 
already  found.  We  must  always  endeavour,  therefore,  to 
arrive  at  such  a  form,  however  complicated  the  equation  may 
be  when  first  formed.  We  shall  give,  in  the  course  of  this 
section,  the  rules  which  serve  to  facilitate  these  reductions. 

493.  Let  us  begin  with  the  simplest  cases,  and  suppose,  first, 
that  we  have  arrived  at  the  equation  a;  -f  9  =  16  ;  we  see  imme- 
diately that  X  =z  7,  And,  in  general,  if  we  have  found  x  +  a 
=  bf  where  a  and  b  express  any  known  numbers,  we  have  only 


160  Algebra.  Sect-4V 

to  subtract  a  from  both  sides,  to  obtain  the  equation  x  =  h  —  a, 
>vhich  indicates  the  value  of  x. 

494.  If  the  equation  which  we  have  found  is  a?  —  a-=}),  w^e 
add  a  to  both  sides,  and  obtain  the  vahie  of  a;  =  6  +  a.    ' 

We  proceed  in  the  same  manner,  if  the  equation  has  this 
form,  X  —  a  =  aa  +  1 ;  for  we  shall  have  immediately  a?  =  aa 

+  «  +  !. 

In  this  equation,  x  —  8a  =  20  —  Qa,  we  find  ^  =  20  ^ — ^  6fl 
-f-  8a,  or  ^  =  20  +  2a. 

And  in  this,  ^  +  6a  =  20  +  Sa,  we  have  a:  =  20  +  3a  —  6a, 
or  X  =  W  —  3a. 

495.  If  the  original  equation  has  this  form,  x  —  a  -^-b  =:  c, 
we  may  begin  by  adding  a  to  both  sides,  which  will  give  x  -f- 
h  =  c  -j-  a;  and  then  subtracting  b  from  both  sides,  we  shall 
find  X  =  c  -\-  a  —  b.  But  we  might  also  add  -fa  — ^  6  at  once 
to  both  sides  ;  by  this  we  obtain  immediately  a;  =  c  -f  a  —  b. 

So  in  the  following  examples, 

If  a?  —  2a  +  3&  =  0,  we  have  x  =  9.a  —  36. 

\ix  —  Sa  -f  26  =  25  +  a  +  26,  we  have  a?  =  25  +  4a. 

If  X  —  9    -f  6a  =  25  +  2a,  we  have  a?  =  34  —  4a. 

496.  When  the  equation  which  we  have  found  has  the  form 

ax  =  6,  we  only  divide  the  two  sides  by  a,  and  we  have  x  =  —, 

a 

But  if  the  equation  has  the  form  ax  +  b  —  c  =  d,  we  must  first 

make  the  terms  tliat  accompany  ax  vanish,  by  adding  to  both 

sides  —  6  -f  c  I  and  then  dividing  the  new  equation,  ax  =  d  — 

^  +  c,  by  a,  we  shall  have  x  =  . 

We  should  have  found  the  same  value  by  subtracting  -f  6  —  c 
from  the  given  equation ;  that  is,  we  should  have  had,  in  the 

same  form,  ax  =  d  —  6  +  c,  and  x  =  .    Hence, 

If  2a;  +  5  =  17,  we  have  2x  =  12,  and  x  =  6, 
If  So:  —  8  =  7,  we  have  3x  =  15,  and  x  =  5, 
If  4^  —  5  —  Sa  =  15  -f  9a,  we  have  4a;  =  20  -f-  12a,  and, 
consequently,  x  =  5  -\-  3a, 

497.  When  the  first  equation  has  the  form  —  =  6,  we  multiply 
both  sides  by  a,  in  order  to  have  x  =  a6. 


Chap.  2.  Of  Compound  Quantities,  16,1 

But  if  we  have  '—  +6  —  c  =  d,  we  must  first  make  —  =  a 
a  ^  a 

—  6  +  c ;    after  which,  we  find  a?  =(d  —  b  +c)  a  z^ad-^ 

ab  +  ac. 

Let  i  X  —  3  =  4,  we  have  |  a?  =  7,  and  x  =  14. 

Let  ^x  —  1  +  2a  =  3  +  a,  we  have  j  a?  =  4  —  a,  and  x  = 
12  —  3a. 

Let 1  =  a,  we  have —  =  a+  1,  and  x  =  aa —  1. 

a — 1  a — 1 

498.  When  we  have  arrived  at  such  an  equation  as  y-  =  c, 
we  first  multiply  by  b,  in  order  to  have  ax  =  be,  and  then  divid- 
ing by  a,  we  find  x  =  —, 

If  —  —  c  =  (f,  we  begin  by  giving  the  equation  this  form  -r 
=  d  -f  c,  after  which,  w^e  obtain  the  value  of  ax  =zbd  -{-  be,  and 

tbatofx  =  *±!±. 
a 

Let  us  suppose  ^  x  —  4  =  1,  we  shall  have  ^  x  =  5,  and  2x 
=  15  ;  wherefore  x  =  y ,  or  7i. 

U  ^  X  -^  ^  =z  5,  we  have  |ir=:5— i  =  |;  wherefore  5x 
=  18,  and  x  =  6. 

499.  Let  us  now  consider  the  case,  whicli  may  frequently 
occur,  in  which  two,  or  more  terms  contain  the  letter  x,  either 
on  one  side  of  the  equation,  or  on  both. 

If  those  terms  are  all  on  the  same  side,  as  in  the  equation  x  -f 
^x-f  5  =  11,  we  havex  -^^  a?  =  6,  or  3a;  =  12,  and,  lastly, a: =4. 

Let  X  +  1  X  +  i  =  44,  and  let  the  value  of  x  be  recjuired  : 
if  we  first  multiply  by  3,  we  have  4x  -f  |  x  =  132  ,•  then  multi- 
plying by  2,  we  have  llx  =  264  ;  wherefore  x  =  24.  We  might 
have  proceeded  more  shortly,  beginning  with  the  reduction  of 
the  three  terms  which  contain  x,  to  the  single  term  y  x  ;  and 
then  dividing  the  equation  y  x  =  44,  by  11,  we  should  have 
had  J  X  =  4,  wherefore  x  =  24. 

Let  I  X  —  ^  X  -{-1  X  =  If  we  shall  have,  by  reduction,  ^^^  <^ 
=  1,  and  X  =  2|. 

.21 


16iJ  nMlgm  Algebra.  Sect.  4. 

Let,  mw'e  generally,  ax  —  bx -j-  ex  =  d ;  this  is  tlie  same  as 
(a  —  b  -}-  c)  X  z=d9  whence  we  derive  x  = 


a — 6+c  * 

500.  "When  there  are  terms  containing  x  on  both  sides  of  the 
equation,  we  begin  by  making  such  terms  vanish  from  the  side 
from  which  it  is  most  easily  done ;  that  is  to  say,  in  which 
there  are  fewest  of  them. 

If  we  have,  for  example,  the  equation  5x  +2  =  a^  +  10,  we 
must  first  subtract  x  from  both  sides,  which  gives  2a?+  2  =  10  ; 
wherefore  2x  =  8,  and  x  =  4, 

Let  ^  -f  4  =  20  —  X ;  it  is  evident  that  2^  +  4  =  20 ;  and 
consequently  2a?  =  16,  and  x  z=  S. 

Let  a;  -f  8  =  32  —  Sx,  we  shall  have  4a;  -f  8  =  32  ;  then  4x 
=  24,  and  x=  6. 

Let  15  —  X  =  20  —- .  Qx,  we  shall  have  15  -f-  x  =  20,  and 
X  =  5. 

Let  1  +  X  =  5  —  I  a?,  we  shall  have  1  -f  |  a?  =  5  ;  after  that 
f  .T  =  4;  3a?  =  8 ;  lastly,  a?  =  |  =  2|. 

If  I  —  ix  =z  1  —  iXf  we  must  add  ^x,  which  gives  ^  =  3^ 
-f.  j\x;  subtracting  -i,  there  remains  y^^x  =  i,  and  multiplying 
by  12,  we  obtain  a?  =  2. 

If  1|  —  |.x  =  I  +  |x,  we  add  |a?,  which  gives  11  =  l  -f-  |^a7. 
Subtracting  i,  we  have  1^0:?=  1^,  whence  we  deduce  x  =  1-^^^  = 
^|,  by  multiplying  by  6,  and  dividing  by  7. 

501.  If  we  have  an  equation,  in  which  the  unknown  number 
X  is  a  denominator,  we  must  make  the  fraction  vanish,  by  mul- 
tiplying the  whole  equation  by  that  denominator. 

Suppose  that  we  have  found 8=  12,  we  first  add  8,  and 

a? 

have =  20  ;  then  multiplying  by  x,  we  have  100  =  20a? ; 

and  dividing  by  20,  we  find  x  =  5, 

Let  — —  =  7. 
^ — 1 

If  we  multiply  by  a?  —  1,  we  have  5x  +  3  z=Tx  —  7. 

Subtracting  5x,  there  remains  3  =  Qx  —  7. 

Adding  7,  we  have  2a?  =  10.     Wherefore  x  z=z  5, 


Chap.  3.  Of  Compound  Quantifies.  163 

502.  Sometimes,  also,  radical  signs  are  found  in  equations  of 
the  first  degree.  For  example,  a  number  x  below  100  is  re- 
quired, and  such,  that  the  square  root  of  100  —  x  may  be  equal 
to  8,  or  v/  (100  —  x)  =  S  ;  the  square  of  both  sides  will  be  100 
—  X  =  64,  and  adding  x  we  have  100  =  64  -f-  x;  whence  we 
obtain  or  =  100  —  64  =  36. 

Or,  since  100  —  a?  =  64,  we  might  have  subtracted  100  from 
both  sides  ;  and  we  should  then  have  had  —  x  =  —  36  ;  whence 
multiplying  by  —  1,  x  =  36. 


CHAPTER  III. 

Of  the  Solution  of  Questions  relating  to  the  preceding  chapter, 

503.  ^lestion  I.  To  divide  7  into  two  such  parts,  that  the 
greater  may  exceed  the  less  by  3. 

Let  the  greater  part  =  x,  the  less  will  be  =  7  —  x ;  so  that 
X  =  7  —  X  -}-  3,  or  X  =  10  —  X  ;  adding  x,  we  have  2x  =  10  ; 
and,  dividing  by  2,  the  result  is  ^t;  =  5. 

Answer.     The  greater  part  is  therefore  5,  and  the  less  is  2. 

Question  II.  It  is  required  to  divide  a  into  two  parts,  so  that 
the  greater  may  exceed  the  less  by  b. 

Let  the  greater  part  =  x,  the  other  will  be  a  —  x ;  so  that 
X  :=.  a  —  X  +  h  ;  adding  x,  we  have  2.x  =  a+h;  and  dividing 

by  2,  0;=  -i-. 

Another  Solution.  Let  the  greater  part  =  x ;  which  as  it  ex- 
ceeds the  less  by  &,  it  is  evident  that  the  less  is  smaller  than  the 
other  by  6,  and  therefore  must  be  =  x  —  h.  Now  these  two 
parts,  taken  together,  ought  to  make  a  ;  so  that  2x  —  bz=  a; 

adding  b,  we  have  Qx  =  a  -f-  6,  wherefore  x  =  —^ ,  which  is 

At 

the  value  of  the  greater  part ;  that  of  the  less  will  be  -—  —  &, 

jt 

a-\-h         2&         a — b 
or — ,  or  .  ( 

2  2  2 

504.  Question  III.  A  father  who  has  three  sons,  leaves  them 
1600  crowns.    The  will  specifics,  that  the  eldest  shall  have  200 


164  Algebra,  Sect.  A. 

crowns  more  than  the  second,  and  that  the  second  shall  have 
100  crowns  more  the  youngest.     Required,  the  share  of  each  ? 

Let  the  share  of  the  tliird  son  =  x ;  then,  that  of  the  second 
will  be  =  0?  +  100,  and  that  of  the  first  =  a?  +  SOO.  Now, 
these  three  shares  make  up  together  1600  crowns.  We  have, 
therefore, 

^x  -f  400  =  1600 
3x  =  1200 
and  X  =    400. 
Answei\     The  share  of  the  youngest  is  400  crowns  ;  that  of 
the  second  is  500  crowns  ;  and  that  of  the  eldest  is  700  crowns. 

505.  Question  IV.  A  father  leaves  four  sons,  and  8600  livres  ; 
according  to  the  will,  the  share  of  the  eldest  is  to  be  double  that 
of  the  second,  minus  100  livres  ^  the  second  is  to  receive  three 
times  as  much  as  the  thiid,  minus  200  livres  ;  and  the  third  is 
to  receive  four  times  as  much  as  the  fourth,  minus  300  livres. 
Required,  the  respective  portions  of  these  four  sons  ? 

Let  us  call  x  the  portion  of  the  youngest ;  that  of  the  third 
son  will  be  =  407  —  300 ;  that  of  the  second  =12x  —  1100,  and 
that  of  the  eldest  =  24x  —  2300.  The  sum  of  these  four  shares 
must  make  8600  livres.  We  have,  therefore,  the  equation  41a; 
—  3700  =  8600,  or  41  a;  =  l-SSOO,  and  x  =  300. 

Answer.  The  youngest  must  have  300  livres,  the  third  son 
900,  the  second  ^500,  and  the  eldest  4900. 

506.  question  V.  A  man  leaves  11000  crowns  to  be  divided 
between  his  widow,  two  sons,  and  three  daughters.  He  intends 
that  the  mother  should  receive  twice  the  share  of  a  son,  and  each 
son  to  receive  twice  as  much  as  a  daughter.  Required,  how 
much  each  of  them  is  to  receive  ? 

Suppose  the  share  of  a  daughter  =  a?,  that  of  a  son  is  conse- 
quently =  2x,  and  that  of  the  widows  4a? ;  the  whole  inheritance 
is  therefore  307  +  4a^  +  4a^ ;  so  that  Ux  =  11000,  and  x  =  1000. 

Answer,     Each  daughter  receives       1000  crowns, 

So  that  the  three  receive  in  all  3000 

Each  son  receives  2000  crowns. 

So  that  both  the  sons  receive  4000 

And  the  mother  receives  .         4000 

Sum  11000  crowns. 


Chap.  S.  Of  Compound  ^lantities.  165 

507.  Question  VI.  A  father  intends,  by  bis  will,  that  his 
three  sons  should  share  his  property  in  the  following  manner ; 
the  eldest  is  to  receive  1000  crowns  less  than  half  the  whole  for- 
tune ;  tlie  second  is  to  receive  800  crowns  less  than  the  third  of 
the  whole  property  ;  and  the  third  is  to  have  600  crowns  less 
than  the  fourth  of  the  property.  Required,  the  sum  of  the  whole 
fortune,  and  the  portion  of  each  son  ? 

Let  us  express  the  fortune  by  x. 

The  share  of  the  first  son  is  ^x  —  1000 

That  of  the  second  i'^  — -    800 

That  of  the  third  io;  —    600. 

So  that  the  three  sons  receive  iji  all  ^x  -{-  ^x  -\-  ^x  —  2400, 
and  this  sum  must  be  equal  to  x. 
We  have,  therefore,  the  equation  i|ct?  —  2400  =  x. 
Subtracting  x,  there  remains  -j\x  —  2400  =  0. 
Adding  2400,  we  have  -^^x  =  2400.     Lastly  multiplying  by 
12,  the  product  is  x  equal  to  28800. 

Jinswer,     The  fortune  consists  of  28800  crowns,  and 

The  eldest  of  the  sons  receives  13400  crowns 

The  second  8800 

The  youngest  6600 

28800  crowns. 

508.  Question  VII.  A  father  leaves  four  sons,  who  share  his 
property  in  the  following  manner : 

The  first  takes  the  half  of  the  fortune,  minus  3000  livres. 

The  second  takes  the  third,  minus  lOOu  livres. 

The  third  takes  exactly  the  fourth  of  the  property. 

The  fourth  takes  600  livres,  and  the  fifth  part  of  the  property. 

What  was  the  whole  fortune,  and  how  much  did  each  son 
receive  ? 

Let  the  whole  fortune  be  =  a? ; 

The  eldest  of  the  sons  will  have  ^x  —  3000 
The  second  ^x  —  1000 

The  third  \x 

The  youngest  ^x  +  600. 

The  four  will  have  received  in  all  ^x  +  \x  +  |^  +  \x  — 
3400,  which  must  be  equal  to  x. 


166  Mgehnu  Sect.  4. 

Whence  results  the  equation  JJo;  —  3400  =  x ; 
Subtracting  Xy  we  have  ^-Lx  —  3400  =  0  ; 
Adding  3400,  we  have  \lx  =  8400  ; 
Dividing  by  17,  we  have  ^^x  =  200  ; 
Multiplying  by  60,  we  have  x  =  12000. 
Answer,     The  fortune  consisted  of  12000  livres. 

The  first  son  received  3000 

The  second  3000 

The  third  3000 

The  fourth  3000 

509.  Question  VIII.  To  find  a  number  such,  that  if  we  add 
to  it  its  half,  the  sum  exceeds  60  by  as  much  as  the  number  itself 
is  less  than  65. 

Let  the  number  =  x,  then  x  +  ^x  —  60  =  65  —  x;  that  is 
to  say  ^x  —  60  =  65  —  x^ 

Adding  x,  we  have  f.x  —  60  =  65  ; 
Adding  60,  we  have  fa?  =  125 ; 
Dividing  by  5,  we  have  i^c  =  25  ; 
Multiplying  by  2,  we  have  x  =  50. 
Answer,    The  number  sought  is  50. 

510.  Question  IX.  To  divide  32  into  two  such  parts,  that  if 
the  less  be  divided  by  6,  and  the  greater  by  5,  the  two  quotients 
taken  together  may  make  6. 

Let  the  less  of  the  two  parts  sought  z=  x ;  the  greater  will  be 

X 

=  32  —  x;  the  first,  divided  by  6,  gives  — ;  the  second,  divid- 
ed by  5,  gives  ^         ;  now,  —  +  ^  7"    =  6.     Sc^  that  multiply- 

D  O  0 

ing  by  5,  we  have  |a;  -f  32  —  a?  =  30,  or  —  Jo?  -f  32  =  30. 
Adding  |x,  we  have  32  =  30  +  ^x. 
Subtracting  30,  there  remains  2  =  ^x. 
Multiplying  by  6,  we  have  a  =  12. 
Answer.   The  two  parts  are  ;  the  less  =  12,  the  greater  =  20. 

511.  Question  H,  To  find  such  a  number  that  if  multiplied 
by  5,  the  product  shall  be  as  much  less  than  40,  as  the  number 
itself  is  less  than  12. 

Let  us  call  this  number  x.  It  is  less  than  12  by  12  — x. 
Taking  tlie  number  x  five  times,  we  have  Sx,  which  is  loss  than 
40  by  40  —  5x,  and  this  quantity  must  be  equal  to  12  —  x. 


Cliap.  S,  Of  Compound  Quantities.  IdT 

We  have  therefore  40  —  5a;  =  12  —  x. 

Adding  5Xf  we  have  40  =  12  4.  4a?. 

Subtractini^  12,  we  have  28  =  4x. 

Dividing  by  4,  we  have  x^:?,  the  number  sought. 

512.  ^lestion  XI.  To  divide  25  into  two  such  parts,  that  the 
greater  may  contain  the  less  49  times. 

Let  tlie  less  part  be  =  x,  then  the  greater  will  be  =  25  —  a?. 
The  latter  divided  by  the  former  ought  to  give  the  quotient  49  ; 

we  have  therefore =49. 

Multiplying  by  x,  we  have  25  —  a?  =  49a?, 

Adding  x,  25  =  50a?. 

And  dividing  by  50  x  =    |. 

Answer.  The  less  of  the  tw^o  numbers  sought  is  ^9  and  the 
greater  is  24| ;  dividing  therefore  the  latter  by  -},  or  multiplying 
by  2,  we  obtain  49. 

5i3.  Question  XII.  To  divide  48  into  nine  parts,  so  that 
every  part  may  be  always  i  gi-eater  than  the  part  which  pre- 
cedes it. 

Let  the  first  and  least  part  =  a?,  the  second  will  be  =  a?  +  |, 
the  third  =  a;  +  1  &c. 

Now  these  parts  form  an  arithmetical  progression,  whose 
first  term  =  x  ;  therefore  the  ninth  and  last  will  be  =  a?  -f-  4. 
Adding  those  two  terms  together,  we  have  2a?  +  4  ;  multiplying 
this  quantity  by  the  number  of  terms,  or  by  9,  we  have  ISo?  -f 
36 ;  and  dividing  this  product  by  2,  we  obtain  the  sum  of  all 
the  nine  parts  =  9a?  -f  18 ;  which  ought  to  be  equal  to  48.  We 
have,  therefore^  9a?  +  18  =  48  ; 

Subtracting  18,  there  remains  9a?  =  SO. 

And  dividing  by  9,  we  have       x  =  Si. 

Jnswer.  The  first  pait  is  3^,  and  the  nine  parts  succeed  in 
the  following  order : 

123456789 
H  +  3|f  41  +  4  J  +  51  +  5|  -hH  +  6|  +  7f 
AYhich  together  make  48. 

514.  question.  Xlll.  To  find  an  arithmetical  progression, 
whose  first  term  =  5,  last  =10,  and  sum  =  60, 

Here,  we  know  neither  the  difference,  nor  the  number  of 
erms ;  but  we  kn»\v  that  the  first  and  the  last  term  would  cna- 


16i  Algebra,  Sect.  4, 

ble  us  to  express  the  sum  of  the  progression,  provided  only  the 
number  of  terms  was  given.     We  shall,  therefore,  suppose  this 

number  =  oCf  and  express  the  sum  of  the  progression  by  ; 

now  we  know  also  that  this  sum  is  60 ;  so  that  ——  =  60  ;   ioj 

=  4,  and  x  =  S. 

Now,  since  the  number  of  terms  is  8,  if  we  suppose  the  diifer- 
cnce  =  %9  we  have  only  to  seek  for  the  eighth  term  upon  this 
supposition,  and  to  make  it  =  10.     The  second  term  is  5  -f  55, 
the  third  is  5  +  2a,  and  the  eighth  is  5  +  7«  ^  so  tliat 
5  +  7«  =  10 
7z=    5 
and  a  =    4 
Answer,     The  difference  of  the  progression  is  4,  and  the 
number  of  terms  is  8  ,•  consequently  the  progression  is 
12345678 
5  +  5f  +  63  +  7^  +  7f  +  8^  +  9|  +  10, 
The  sum  of  which  =  60. 

515.  Question  XIV.  To  find  such  a  number,  that  if  1  be 
subtracted  from  its  double,  and  the  remainder  be  doubled,  then 
if  2  be  subtracted,  and  the  remainder  divided  by  4,  the  number 
resulting  from  tliese  operations  shall  be  1  less  than  the  number 
sought. 

Suppose  this  number  =  cc  ;  the  double  is  9.x  ;  subtracting  1, 
there  remains  9,x  —  1 ;  doubling  this,  we  have  4x —  2  ;  sub- 
tracting 2,  there  remains  4x  —  4  ;  dividing  by  4,  we  have 
x  —  1 ;  and  this  must  be  one  less  than  x  ;  so  that. 

But  this  is  what  is  called  an  identical  equation  ;  and  shews  that 
X  is  indeterminate ;  or  that  any  number  whatever  may  be  sub- 
stituted for  it. 

516.  Question  XV.  I  bought  some  ells  of  cloth  at  the  rate  of 
7  crowns  for  5  ells,  which  I  sold  again  at  the  rate  of  11  crowns 
for  7  ells,  an-;  I  gained  100  crowns  by  the  traffic.  How  much 
cloth  was  there  ? 

Suppose  that  there  were  x  ells  of  it ;  we  must  first  see  how 
much  the  cloth  cost.  This  is  found  by  the  following  proportion ; 


Chap.  3.  Of  Compound  ^antitits,  169' 

If  five  ells  cost  7  crowns ;  what  do  x  ells  cost  ? 

Answer,  \x  crowns. 

This  was  my  expenditure.  Let  iis  now  see  my  receipt :  we 
must  make  the  following  proportion  ;  as  7  ells  are  to  11  crowns, 
so  are  x  ells  to  y  x  crowns. 

This  receipt  ought  to  exceed  the  expenditure  hy  100  crowns  ; 
we  have,  therefore,  this  equation. 
\^x  =  \x  +  100; 

Subtracting  ^a?,  there  remains  -{-gX  =  100. 

Wherefore  6x  =  3500,  and  x  =  583^. 

Answer.  There  were  583|  ells,  which  were  bought  for  816| 
crowns,  and  sold  again  for  916|  crowns,  by  which  means  the 
profit  was  1 00  crowns. 

517.  Question  XVI.  A  person  buys  12  pieces  of  cloth  for  140 
crowns.  Two  are  white,  three  are  black,  and  seven  are  blue. 
A  piece  of  the  black  cloth  costs  two  crowns  more  than  a  piece  of 
the  white,  and  a  piece  of  blue  clotli  costs  three  crowns  more  than 
a  piece  of  black.     Required  the  price  of  each  kind  ? 

Let  a  white  piece  cost  x  crowns  ;  then  the  two  pieces  of  this 
kind  will  cost  2x,  Further,  a  black  piece  costing  x  +  2,  the 
three  pieces  of  this  colour  will  cost  3x  -f-  6.  Lastly,  a  blue 
piece  costs  x  +  5 ;  wherefore  tlie  seven  blue  pieces  cost  7x  -f 
35.     So  that  the  twelve  pieces  amount  in  all  to  12a;  -f-  41. 

Now,  the  actual  and  known  price  of  these  twelve  pieces  is 
140  crowns  ;  we  have,  therefore,  12x  +  41  =  140,  and  12x  = 
99  ;  wherefore  x  =  S^  ; 

So  that  a  piece  of  white  cloth  costs  81  crowns  ;  apiece  of  black 
cloth  costs  lOi  crowns,  and  a  piece  of  blue  cloth  costs  131  crowns. 

518.  ^estim  X-YIh  A  man  having  bought  some  nutmegs,, 
says  that  three  nuts  cost  as  much  more  than  one  sou  as  four  cost 
him  more  than  ten  liards  :  Required,  the  price  of  those  nuts  ? 

We  shall  call  x  the  excess  of  the  price  of  three  nuts  above  one 

sou,  or  four  liards,  and  shall  say ;    If  three  nuts  cost  x  -\-  4t 

liards,  four  will  cost,  by  the  condition  of  the  question,  a;  +  10 

liards.     Now,  the  price  of  three  nuts  gives  that  of  four  nuts  in 

another  Avay  also,  namely,  by  the  rule  of  three.  We  make  3  :  x 

,   A        A        a  4a-+16 

4-  4  =  4  :     Answer,  i^ — . 

3 


170  Mgebra.  Sect.  4. 

So  that  — ^- —  =  a;  -f  10  ^  or,  4a?  +  16  =  Sac?  +  30 ;  where- 

fore   a?  +  16  =  30 
and    0?  =  14. 

Jnsiver.  Three  nuts  cost  18  liards,  and  four  cost  6  sous; 
wiiercforc  each  cost  6  liards. 

519.  Question  XVIII.  A  certain  person  has  two  silver  cups, 
and  only  one  cover  for  both.  The  first  cup  weighs  12  ounces^ 
and  if  the  cover  be  put  on  it,  it  weighs  twice  as  much  as  tlie 
otiier  cup  ;  but  if  the  other  cup  be  covered,  it  weighs  three  times 
as  much  as  the  first :  Required,  the  weight  of  the  second  cup 
and  that  of  the  cover  ? 

Suppose  the  weight  of  the  cover  =  x  ounces  ;  the  first  cup 
being  covered  will  weigh  x  -f  12  ounces.  Now  this  weight 
being  double  that  of  the  second  cup,  this  cup  must  weigh  ^x  +6. 
If  it  be  covered,  it  will  weigh  |x  -f  6  ;  and  this  weight  ought  to 
be  the  triple  of  12,  that  is,  three  times  the  weight  of  the  first 
cup.  We  shall  therefore  have  the  equation  |a?  -f  6  =  36,  or  |x 
=  30  ;  wherefore  |x  =  10  and  x  =  20. 

Answer,  The  cover  weighs  20  ounces,  and  the  second  cup 
weighs  16  ounces. 

520.  Question  XIX.  A  banker  has  two  kinds  of  change  ; 
there  must  be  a  pieces  of  the  first  to  make  a  crown ;  and  there 
must  be  h  pieces  of  the  second  to  make  the  same  sum.  A  per- 
son wishes  to  have  c  pieces  for  a  crown  ;  how  many  pieces  of 
each  kind  must  the  banker  give  him  ? 

Suppose  the  banker  gives  x  pieces  of  the  first  kind  ;  it  is  evi- 
dent that  he  will  give  c  —  x  pieces  of  the  other  kind.  Now,  the  x 

pieces  of  the  first  are  worth  —  crown,  by  the  proportion  a  :  1  = 

X  :  —  ;  and  the  c  —  x  pieces  of  the  second  kind  are  worth    

a  CL 

'^'  C      JC  oc 

crown,  because  we  have  h  :  1  =  c  —  x  i  —r-  .     So  that, f- 

iri^  =;:  X  ;  or  ^-^  -f  c  — ■  ^  =  6 ;  or  hx  -^  ac  —ax  =  ah  ;  or, 

ih — m 


1  1  f^^ — ^^ 

rather,  ox  —  ax  z=z  ab  —  ac  ;  whence  we  have  x  =  -7- —  9  01* 


Chap.  S.,  Of  Compound  ^uintities.  171 

rt(6— c)        ^  ^1       ^        ^       bc—ab       b{c'^a) 

X  =  ^_I.     Consequently,  ^  -  a:  =  -^_-  =  -^A 

Jinswer,  The  hanker  will  give  ^^.^^^  pieces  of  the  first  kind. 

and     -      ^'    pieces  of  the  second  kind. 

b — a 

Remark,  These  two  nuinhers  are  easily  found  by  the  inile  of 
three,  when  it  is  required  to  apply  the  results  which  we  have 

obtained.  To  find  the  first  we  say  5  h  —  a-.h  —  cz=.a:  "^^        \ 

The  second  number  is  found  thus  ;  b  —  a  :  c  —  a=zb  :     \       K 

b — a 

It  ought  to  be  observed  also,  that  a  is  less  than  6,  and  that  c 

is  also  less  than  b  ;  but  at  the  same  time  greater  than  a,  as  the 

nature  of  the  thing  requires. 

521.  ^lestion  XX.  A  banker  has  two  kinds  of  change  ;  10 
pieces  of  one  make  a  crown,  and  20  pieces  of  the  other  make  a 
crown.  Now,  a  person  wishes  to  change  a  crown  into  If 
pieces  of  money  :  How  many  of  each  must  he  have  ? 

We  have  here  a  =  10,  6  =  20,  and  c  =  17 ;  which  furnishes 
the  following  proportions ; 

I.  10  :  3  =  10  :  3,  so  that  the  number  af  pieces  of  the  first 
kind  is  3. 

II.  10  :  7  =  20  :  14,  and  there  are  14  pieces  of  the  second 
kind. 

522.  ^lestion  XXI.  A  father  leaves  at  his  death  several 
children,  who  share  his  property  in  the  following  manner ; 

The  first  receives  a  hundred  crowns  and  the  tenth  part  of  tlie 
remainder. 

Tlie  second  receives  two  hundred  crowns  and  the  tenth  part 
of  what  remains. 

The  third  takes  three  hundred  crowns  and  the  tenth  part  of 
what  remains. 

The  fourth  takes  four  hundred  crowns  and  the  tenth  part  of 
what  then  remains,  and  so  on. 

Now  it  is  found  at  the  end,  that  tlie  property  has  been  divid- 
ed equally  among  all  the  children.  Required,  bow  much  it  was, 
how  many  children  there  were,  and  how  much  each  received  ? 


172 


Mgebru, 


8ect  4. 


This  question  is  riiiher  of  a  singular  nature,  and  therefore 
deserres  particular  attention.  In  order  to  resolve  it  more  easily, 
we  shall  suppose  the  whole  fortune  =  «  crowns  ;  and  since  all 
the  children  receive  the  same  sum,  let  the  share  of  each  =  x,  by 

which  means  the  numher  of  children  is  expressed  by  —.    This 

being  laid  down,  we  may  proceed  to  the  solution  of  the  question, 
which  w  ill  be  as  follows. 


Sum,  or  pro- 
perty to  be 
divided. 

Order  of 

the 
Children. 

/J 

ist. 

X  X 

2d. 

X>  —  2x 

3d. 

a  —  3x 

4th. 

%  —  4a? 

5th. 

%  —  5x 

gth. 

Portion  of  each. 


V 


X  =  100  + 

X  =  200  -f 
X  =  300  + 
X  =  400  -f- 
X  =  500  -f 
X  =  600  + 


V 

z— 100 

To 

2^—^—200 

10 
z — 2.^—300 


iO 

z—S.v—iOO 


'  10 
;jp— 4.r — 500 


10 
z — 5.x — 600 

To 


Differences. 


100 
100 
100 
100 


■v 


07—100 

To 

^—100 

Tu 

^—100 

10 
^ — 100 


10 


=  0 
=  0 
=  0 
=  0 


and  so  on. 


Wc  have  inserted,  in  the  last  column,  the  differences  which 
we  obtain  hy  subtracting  each  portion  from  that  which  follows. 
Now  all  the  portions  being  equal,  each  of  the  differences  must 
be  =  0.  And  as  it  happens  that  all  these  differences  are  express- 
ed exactly  alike,  it  will  be  sufficient  to  make  one  of  them  equal 

to  nothing,  and  we  shall  have  the  equation  100 — =  0. 

Multiplying  by  10,  we  have  1000  —  x  —  100  =  0,  or  900  —  x 
=  0  ;  consequently  a;  =  900. 

AYe  now  know,  therefore,  that  the  share  of  each  child  was 
900  crowns ;  so  that  taking  any  one  of  the  equations  of  the 
third  column,  the  first,  for  example,  it  becomes,  by  substituting 

^      100 

the  value  of  x,  900  =  100-1 — ,  whence  we  immediately 

obtain  the  value  of  % ;  for  we  have  9000  =  1000  -f  « —  100,  or 
9000  =  900  -f  % }  wherefore  s;  =  8100  ;  and  consequently  —  =  9. 


Chap.  4.  Of  tovipound  ^antities,  17S 

Msnver,  So  that  the  number  of  children  =  9 ;  the  fortune 
left  by  the  father  =  8100  crowns;  and  the  share  of  each  child 
=  900  crowns. 


CHAPTER  IV. 

Of  the  Resolutions  of  two  m-  more  Equations  of  the  First  Degree, 

5^S,  It  frequently  happens  that  we  are  obli<*ed  to  introduce 
into  algebraic  calculations  two  or  more  unknown  quantities, 
represented  by  the  letters  x,  y,  % ;  and  if  the  question  is  deter- 
minate, we  are  brought  to  the  same  number  of  equations  ;  from 
which,  it  is  then  required,  to  deduce  tiic  unknown  quantities. 
As  we  consider,  at  present,  those  equations  only,  whi(  h  contain 
no  powers  of  an  unknown  quantity  higher  than  the  first,  and  no 
products  of  two,  or  more  unknown  quantities,  it  is  evident  that 
these  equations  will  all  have  the  form  a%  +  by  -^  cxz=  d, 

524.  Beginning  therefore  with  two  equations,  we  shall  en- 
deavour to  find  from  them  the  values  of  x  and  y.  That  we  may 
consider  this  case  in  a  general  manner,  let  the  two  equations  be, 
I.  ax  -\-  by  =  c,  and  II.  fx-}-gy  =  h,  in  which  a,  6,  c,  and  /,  g, 
h  are  known  numbers.  It  is  required,  therefore,  to  obtain,  from 
these  two  equations,  the  two  unknown  quantities  x  and  y, 

525.  The  most  natural  method  of  proceeding  will  readily 
present  itself  to  the  mind ;  which  is  to  determine,  from  both 
equations,  the  value  of  one  of  the  unknown  quantities,  x  for 
example,  and  to  consider  the  equality  of  those  two  values ;  for 
then  we  shall  have  an  equation,  in  which  the  unknown  quantity 
7/  will  be  found  by  itself,  and  may  be  determined  by  the  rules 
which  we  have  already  given.  Knowing  t/,  we  have  only  to 
substitute  its  value  in  one  of  the  quantities  that  express  x, 

526.  According  to  this  rule,  we  obtain  from  the  first  equa- 
tion, X  =    "^  '^,  and  from  the  second,  x  =    ~"^^  ;  stating 

these  two  values  equal  to  one  another,  we  have  this  new  equa- 
tion : 

c — by       h — gy 


174  AlgehHi.  Sect.  4. 

multiplying  by  a,  the  pi-oduct  is  c  —  hyz=  "  ^"^"^V  .  multiplying 

by/,  the  product  is/c  — Jby  =  a/i  —  agy  ;  adding  agy,  we  have 
Jc  — fhy  -f  agy  =  a/i ;  subtracting  /c,  there  renjains  — fhy  + 
agy  =z  ah  — fc ;  or  (ag  —  If)  y  z=i  ah  — /c ;  lastly,  dividing  by 

ag  —  hf,  we  have  w  =  ^  ^~"-^^  . 

In  order  now  to  substitute  this  value  of  y  in  one  of  the  two 
values  which  we  have  found  of  x,  as  in  the  first,  where  x  = 

^"^  ^9  we  shall  first  have  —  hv  =  —  ^        //  ;  whence  c  —  hit 

__  ahh-\-hcf      ^  _  acg — bcf — abh-\-hcf      acg — abh  ^ 

and  dividing  hy  a,  x  =  — -^  =  -^ ^. 

a  ag^bf 

527.  Question  I.  To  illustrate  this  method  by  examples,  let 
it  be  proposed  to  find  two  numbers,  whose  sum  may  be  =  15, 
and  difference  =  7. 

Let  us  call  the  greater  number  x,  and  the  less  y.  We  shall 
have, 

h  X  -^y  =  15,  and  II.  a^  —  V  =^7. 

The  first  equation  gives  x  =  15  —  y,  and  the  second  x  =z  7 
-f-  y;  whence  results  the  new  equation  15  —  y  =  7  -j-  y.  So 
that  15  =  7  +  2y ;  2y  =8,  and  y  =  4  ;  by  which  means  we 
find  ^  =  11. 

Jlnswer,    The  less  number  is  4,  and  the  greater  is  11. 

528.  Question  II.  We  may  also  generalize  the  preceding 
question,  by  requiring  two  numbers,  whose  sum  may  be  =  a, 
and  the  difference  =  b. 

Let  the  greater  of  the  two  be  =  x,  and  the  less  =  y. 

We  shall  have  I.  x  -j- y  :=  a^  and  II.  it?  —  !/  =  ^  ?  the  first 

equation  gives  x  =  a  —  y  ;  and  the  second  x  =  b  -^y. 

Wherefore  a  —  y  =  b  -^  y  ;  a=  b  -{- 2y ;  2y=za  —  b;  lastly, 

fl— &         ,                     ,,                                       a-\-b      a-\-b 
y  =   ,  and  consequently  x  =  a  — >  y  z^a —.  =  — — -. 

Answer.    The  greater  number,  or  x,  is  =  — - ,  and  the  less, 

or  1/,  is  =   —— ,  or  which  comes  to  the  same,  a;  =  |a  -|- 15,  and 


Chap.  4.  Of  Compound  Quantities,  175 

'tf  =  |fl  —  \h ;  and  hence  we  derive  the  following  theorem. 
TV  hen  the  sum  of  any  two  numbers  is  a,  and  tlieir  difference  is  b, 
ihe  greater  of  the  two  numbers  will  be  equal  to  half  the  sum  plus 
fiaJf  the  difference  ;  and  the  less  of  tJie  two  numbers  will  be  equal  to 
lialf  the  sum  minus  haff  the  difference. 

529.  We  may  also  resolve  the  same  question  in  the  following 
manner  ; 

Since  the  two  equations  are,  oc  -j-  y  =  a,  and  x  —  y  =z  b  ;  if 
we  add  one  to  the  other,  we  have  2x  =z  a  -j-  b. 

Wherefore  x  =  . 

2 

Lastly,  subtracting  the  same  equation  from  the  other,  we  have 
2i/  =  a  —  b;  wherefore  y  =  -^  . 

530.  Question  III.  A  mule  and  an  ass  were  carrying  burdens 
amounting  to  some  hundred  weight.  The  ass  complained  of  his, 
and  said  to  the  mule,  I  need  only  one  hundred  weight  of  your 
load,  to  make  mine  twice  as  heavy  as  yours.  The  mule  answer- 
ed, Yes,  but  if  you  gave  me  a  hundred  weight  of  yours,  I  should 
be  loaded  three  times  as  much  as  you  would  be.  How  many 
hundred  weight  did  each  carry  ? 

Suppose  the  mule^s  load  to  be  x  hundred  weight,  and  that  of 
the  ass  to  be  y  hundred  weight.  If  the  mule  gives  one  hundred 
w  ciglit  to  the  ass,  tlie  one  will  have  «/  -f  1,  and  there  will  remain 
for  the  other  x  —  1  ;  and  since,  in  this  case,  the  ass  is  loaded 
twice  as  much  as  the  mule,  we  have  y  •}-  1  =  9.x  —  2. 

Further,  if  the  ass  gives  a  hundred  weight  to  the  mule,  the 
latter  has  a?  -f  1,  and  the  ass  retains  y  —  1 ;  but  the  burden  of 
the  former  being  now  three  times  that  of  the  latter,  we  have 
a?  +  1  =  31/  —  3. 

Our  two  equations  will  consequently  be, 

1. 1/  4-  1  =  2x  —  2,  II.  X  -i-l  =  5y  —  3. 

The  first  gives  x  =  -^^ ,  and  the  second  gives  x  =:  5y  —  4  ; 

whence  we  have  the  new  equation  ^i-  =3^  —  4,  which  gives 

y  =z  y ,  and  also  determines  the  value  of  x,  which  becomes  2|. 
Jnswer.     The  mule  carried  2|  hundred  weight,  and  the  ass 
carried  2i  hundred  weight. 


176-  .  Mgeh-a.  Sfect.  4. 

551.  When  there  are  three  unknown  numbers,  and  as  many 
equations  ;  as,  for  example,  L  x  -{-  y  —  ®  =  8,  II.  x  -|-  a  —  y 
=  9,  III.  1/  +  a  —  ;r  =  10,  we  begin,  as  before,  by  deducing  a 
value  of  c€  from  each,  and  we  Iiave,  from  the  P%  a?  =  8  -f  a  —  y- 
from  the  11^,  a'  =  9  +  y  —  a  ;  and  from  the  III^,  x  =  y  ^  ^ 
--10. 

,  Comparing  the  first  of  these  values  with  the  second,  and  after 
that  with  the  third  also,  we  have  the  following  equations  ; 

I.  8-f«  —  y  =:  9  +y  —  «,    II.  8+®  —  y  z=y  ^%  —  10. 

Now,  the  first  gives  22; —  2^  =  1>  and  the  second  gives  2i/  = 
18,  or  7/  =  9  ;  if  therefore  we  substitute  this  value  of  2/  in  9.%  — 
%  =  1,  we  have  £2^  —  18  =  1,  and  2;i  =  19,  so  that  a  =  9|  ; 
it  remains  therefore  only  to  determine  a?,  which  is  easily  found 

Here  it  happens,  that  the  letter  «  vanishes  in  tlie  last  equation, 
and  that  the  value  of  y  is  found  immediately.  If  this  had  not 
been  the  case,  we  should  have  had  two  equations  between  a  and 
y,  to  be  resolved  by  the  preceding  rule. 

532.  Suppose  we  had  found  the  three  following  equations. 
I.  Sx  -{-  5y  —  4%;  =  25,  II.  5x  —  2t/  -f  3»  =  46, 
III.  32/  -f  52i  —  a;  =  62. 
If  we  deduce  from  each  the  value  of  Xy  we  shall  have 

I.  X  = f^ — ,  II.  x  =  — -3-| 

III.  ir  =  3t/  +  5«  — .  62. 

Comparing  these  three  values  together,  and  first  the  third 

25     5v  -f-  Az 
with  the  first,  we  have  Si/  -f-  5«  —  62  =    ^-^ —  ;    multi- 

plying  by  3,  9y  -^  15%  ■ —  186  =  25  —  5y  -j- 4%  ;  so  that  9y  + 
15%  =  211  —  5y  -j.  4%,  and  14y  -f-  11a  =  211  by  the  first  and 
tlie  third.     Comparing  also  the  third  with  the  second,  we  have 

3^  4.  52i  —  62  =  I5i^^n2f,  or  46  +  2y -— S%  z=z  15y  +  25z 

—  310,  which  when  reduced  is  356  =  151/  +  QSz. 

We  shall  now  deduce,  from  these  two  new  equati6ns,  the  value 
of  ^; 

I.  211  =  Hz/  -f.  11«  ;  wherefore  I4y  =  211  -—  ll2^,  and  y  = 
211 — 11% 


Chap.  4.  Of  Compound  ^lantities,  177 

11.  356=   13^+  28»;  wherefore  13i/  =  356  —  28«,  and 

356—282: 

^=—13— 

mu       ^           1       ^        XI                     r      211— U;:;       356—28;:; 
These  two  values  form  the  new  equation ; = — . 

which  becomes,  2743  —  143«  =  4984  —  392«,  or  249»  =  2241, 
whence  «  =  9. 

This  value  being  substituted  in  one  of  the  two  equations  of  y 
and  »,  we  find  y=  8 ;  and  lastly  a  similar  substitution  in  one  of 
the  three  values  of  x,  will  give  a?  =  7. 

533,  If  there  were  more  than  three  unknown  quantities  to  be 
determined,  and  as  many  equations  to  be  resolved,  we  should 
proceed  in  the  same  manner ;  but  the  calculations  would  often 
prove  very  tedious.  ' 

It  is  proper,  therefore,  to  remark,  that,  in  each  particular 
case,  means  may  always  be  discovered  of  greatly  facilitating  its 
resolution.  Tiiese  means  consist  in  introducing  into  the  calcu- 
lation, beside  the  principal  unknown  quantities,  a  new  unknown 
quantity  arbitrarily  assumed,  such  as,  for  example,  the  sum  of 
all  tlie  rest ;  and  when  a  person  is  a  little  practised  in  such  cal- 
culations he  easily  perceives  what  it  is  most  proper  to  do.  The 
following  examples  may  serve  to  facilitate  the  application  of 
these  artifices. 

534.  QiiestionTV,  Three  persons  play  together ;  in  the  first 
game,  the  first  loses  to  each  of  the  other  two,  as  much  money 
as  each  of  tliem  has.  In  the  next,  the  second  person  loses  to 
each  of  the  other  two,  as  much  money  as  they  have  already. 
Lastly,  in  the  third  game,  the  first  and  the  second  person  gain 
each,  from  the  third,  as  much  money  as  tliey  had  before.  Tliey 
then  leave  off,  and  find  tliat  they  have  all  an  equal  sum,  namely 
24  louis  each.  Required,  with  how  much  money  each  sat  down 
to  play  ? 

Suppose  that  the  stake  of  the  first  person  was  x  louis,  that  of 
the  second  i/,  and  that  of  the  third  a.  Further,  let  us  make  the 
sum  of  all  the  stakes,  or  x  -\-  y  -\-  a,  =  s.  Now,  the  first  person 
losing  in  the  first  game  as  much  money  as  the  other  two  have, 
he  loses  s  —  x ;  (for  he  himself  having  had  x,  the  two  others 
23 


1^  JUgdn-a,  Sect.  4. 

must  have  bad  s  —  x) -,  wherefore  there  will  remain  to  him  9.x 
—  s ;  tlie  second  will  have  2?/,  and  the  third  will  have  2x>. 

So  that,  after  the  first  game,  each  will  have  as  follows ; 
the  I.  Qx  —  5,  the  II.  Qtj,  the  III.  2». 

In  the  second  game,  tlie  second  pei'son,  who  has  now  2i/,  loses 
as  much  money  as  the  other  two  have,  that  is  to  say  s  —  2?/ ;  so 
that  he  has  left  4^  —  s.  With  regard  to  the  others,  they  will 
each  have  double  what  they  had ;  so  that  after  the  second  game^ 
the  three  persons  have ; 

the  I.  4x  —  2s,  the  II.  4?/  —  s,  tlie  IIL  4%. 

In  the  third  game,  the  third  person,  who  has  now  42i,  is  the 
loser ;  he  loses  to  the  first  4x  —  2s,  and  to  the  second  4y  —  s  ', 
consequently  after  this  game  the  three  persons^will  have  ; 
the  I.  Sx  —  4s,  the  II.  Sy  —  2s,  the  III.  Sz  —  s. 

Now,  each  having  at  the  end  of  this  game  24  louis,  we  hav© 
three  equations,  the  first  of  which  immediately  gives  x,  the 
second  i/,  and  the  third  « ;  further,  s  is  known  to  he  =  72,  since 
the  three  persons  have  in  all  72  louis  at  the  end  of  the  last  game  ; 
but  it  is  not  necessary  to  attend  to  this  at  first.     We  have 

I.  Sx  —  4s  =  24,  or  8a?  =  24  +  4s,  or  x  =  3  +  -Js  ; 

II.  8^  —  2s  =  24,  or  8i/  =  24  +  2s,  or  i/  =  3  -f  is  5 
III.  Sz  —    s  =  24,  or  82i  =  24  -f  s,  or  »  =  3  +  |s. 

Adding  these  three  values,  we  have 

So  that,  since  x  -^  y  +  x>  =  s,  we  have  s  =  9  -f-  |s  ,*  wherefore 
-1$  =  9,  and  s  =  72. 

If  wc  now  substitute  this  value  of  s  in  the  expressions  which 
we  have  found  for  x,  y,  and  2;,  we  sliall  find  that  before  they 
began  to  play,  the  first  person  had  39  louis ;  the  second  '21  louis ; 
and  the  third  12  louis. 

This  solution  shews,  that  by  means  of  an  expression  for  the 
sum  of  the  three  nnknown  quantities,  we  may  overcome  the 
diHicultics  which  occur  in  the  ordinary  method. 

535.  Although  the  preceding  question  appears  difiicult  at  first, 
it  may  be  resolved  even  without  algebra.  We  have  only  to  try 
to  do  it  imersely.  Since  tlje  players,  when  they  left  off,  had 
each  24  louis,  and,  in  the  third  game,  the  first  and  the  second 
doubled  their  money,  they  must  have  had  before  that  last  game  : 


Chap.  4.  Of  Compmind  ^lantities,  1T9 

The  I.  12,  the  II.  12,  and  the  III.  48. 
In   the  second  game  tlie  first  and  the  third  douhled  their 
money  ;  so  tliat  before  that  game  they  had  ; 

The  I.  6,  the  11.  42,  and  the  III.  24. 
Lastly,  in  the  first  game,  tlie  second  and  the  third  gained 
each  as  much  money  as  they  began  with ;  so  that  at  first  the 
three  persons  had : 

I.  59,   11.21,    III.  12. 
The  same  result  as  we  obtained  by  the  former  solution. 

536.  (lucstion  V.  Two  persons  owe  29  pistoles ;  they  have 
botli  money,  but  neither  of  them  enough  (o  enable  him,  singly  to 
discharge  this  common  debt ;  the  first  debtor  says  therefore  to 
the  second,  if  you  give  me  |  of  your  money,  I  singly  will  imme- 
diately pay  the  debt.  The  second  answers,  that  he  also  could 
discharge  the  debt,  if  the  other  would  give  him  |.  of  his  mohey. 
Required,  how  many  pistoles  each  had  ? 

Suppose  that  the  first  has  x  pistoles,  and  that  the  second  has 
y  pistoles. 

"NVe  shall  first  have,  x  -f-  f?/  =  29  ; 
then  also,  ?/  -|-  |.r  =  29. 
The  first  equation  giVes  x  =  29  —  ^y,  and  the  second,  x  = 

;  so  that  29  —  f !/  =  — ^^^^.      From   this  equation, 

3  3 

w^e  get  y  =  14i  ;  wherefore  x  =  19^. 

Answer,     The  first  debtor  had  19-|  pistoles,  and  the  second 

had  14-|  pistoles. 

537.  Question  VI.  Three  brothers  bought  a  vineyard  for  a 
hundred  louis.  The  youngest  says^  that  he  could  pay  for  it 
alone,  if  the  second  gave  him  half  the  money  which  he  had  ;  the 
second  says,  that  if  the  eldest  would  give  him  only  the  third  of 
his  money,  he  could  pay  for  the  vineyard  singly  ;  lastly,  the 
eldest  asks  only  a  fourth  part  of  the  money  of  the  youngest,  to 
pay  for  the  vineyard  himself.     How  much  money  had  each  ? 

Suppose  the  first  had  x  louis  ;  the  second,  y  louis ;  the  third, 
»  louis  ;  we  shall  then  have  the  three  following  equations  ; 
I.  07  -f  ii/  =  100.         II.  y  ■\.\%=  100. 
III.  %  -f  ix  =  100  ^  two  of  which  only  give  the  value  of  x, 


180  Mgebra,  Sect.  4. 

namely  I.  x  =  100  —  ^r/,  III.  x  =  400  —  42J.  So  that  we 
have  the  equation, 

100  —  iy  =  400  —  4«,  or  4%  —  it(  =  300,  which  must  be 
combined  with  the  second,  in  order  to  determine  y  and  %.  Now, 
the  second  equation  was,  t/  +  -|«  =  100 ;  we  therefore  deduce 
from  it  2/  =  100  —  -Ja ;  and  the  equation  found  last  being  425^ 
—  |i/  =  300,  we  have  y  =.%%  —  600.  Consequently  the  final 
equation  is, 

3  00  —  i»  =  82;  —  600  ;  so  that  8-|z>  =  700,  or  y  z;  =  700, 
and  a  =  84.     Wherefore  2/  =  100  —  ^8  =  72,  and  x  =  64. 

•Answer.  The  youngest  had  64  louis,  the  second  had  72  louis, 
and  the  eldest  had  84  louis. 

538.  As,  in  this  example,  each  equation  contains  only  two 
unknown  quantities,  we  may  obtain  the  solution  required  in  an 
easier  way. 

The  first  equation  gives  y  =  200  —  2a? ;  so  that  y  is  deter- 
mined by  X ;  and  if  we  substitute  this  value  in  the  second  equa- 
tion, we  have  200  — '  9.x  -\-  ^%  z=z  100  5  wherefore  ^z>  =  2x  — 
100,  and  ^  =z  6x  —  300. 

So  that  «  is  also  determined  by  x ;  and  if  we  introduce  this 
value  into  the  third  equation,  we  obtain  6x  —  300  4-  ix  =  100, 
in  which  x  stands  alone,  and  which,  when  reduced  to  25a?  — 
1600  =  0,  gives  x  =  64.  Consequently,  1/  =  200  —  128  =  72, 
and  %  =  384  —  300  =  84. 

539.  We  may  follow  the  same  method,  when  we  have  a  greater 
number  of  equations.  Suppose,  for  example,  that  we  have  in 
general ; 

l,u^—=z  n,  II.  a?  -f-  -|  =  n,    III.  rj  -^ =  w, 

u  0  c 

IV.  «  -f-  -J  =  ?i ;  or,  reducing  the  fractions, 

I.  au  -{-  X  =  an,  II.  hx  -^  y  =  In,  III.  c^  4-  2  =  en, 
IV.  dx>  -^u^  dn. 
Here,  the  first  equation  gives  immediately  x  =  an  —  au, 
and,  this  value  being  substituted  in  the  second,  we  have  abn  — 
ahu  + 1/  =  6/1 ;  so  that  y  =.  bn  —  abn  4-  abu ;  the  substitution  of 
this  value,  in  the  third  equation,  gives  ben  —  abcn  -f  abcu  +«  = 
en ;  wherefore  %  =  en  —  ben  -f-  abcn  —  abm ;  substituting  this 


Chap.  4.  Of  Compmind  ^lantities,  181 

in  the  fourth  equation,  we  Iiave  cdn  —  hcdn  -f  dbcdii  —  ahcdu  + 
u  ■=■  dn.     So  that  dn  —  cdn  -f-  bcdn  —  abcdn  =  —  abcdu  -f  w,  or 
(abed  —  1)  it  =  abcdn  —  bcdn  -f-  cdn  —  dn ;  whence  we  have 
abcdn — bcdn+cdn — dn  (abed — hcd-\-cd — d) 

~"  abed  — I  ~~  abed — I 

Consequently,  we  shall  have, 

abcdn — acdn-j-adn^an  (ahcd-—acd-\-ad — a) 

X  =  : —  n  X   7 — ; ; . 

aoca — 1  abed — 1 

abcdn — abdn-\-abn — bn  {abed — nbd-\-ab — &) 

^  ~~  abed — 1  ~"  ahed — 1 

abcdn — abc7i-\-bcn — en  (abed — nbc4-bc — c) 

%  =  ,      ,       , =71  X  ^^ : J- i. 

abed — 1  aucd — 1 

abcdn — hcdn-\-cdn — dn  {abed — bcd-\-cd — d) 

"~  abed — 1  abed — 1 

540.  Question  VII.  A  captain  has  three  companies,  one  of 
Swiss,  another  of  Swabians,  and  a  third  of  Saxons.  He  wishes 
to  storm  with  part  of  these  troops,  and  he  promises  a  reward  of 
901  crowns,  on  the  following*  condition  ; 

Tliat  each  soldier  of  the  company,  which  assaults,  shall  re- 
ceive 1  crown,  and  that  the  rest  of  the  money  shall  be  equally 
distributed  among  the  two  other  companies. 

Now  it  is  found,  that  if  the  Swiss  make  the  assault,  each  sol- 
dier of  the  other  comjianies  receives  half  a  crown  ;  that,  if  the 
Swabians  assault,  each  of  the  others  receives  j  of  a  crown ; 
lastly,  that  if  the  Saxons  make  the  assault,  each  of  the  others 
receives  ^  of  a  crown.  Required,  the  number  of  men  in  each 
company  ? 

Let  us  suppose  the  number  of  Swiss  =  x,  that  of  Swabians 
=  If,  and  that  of  Saxons  =  %.  And  let  us  also  make  x  + 1/  -f- « 
=  s,  because  it  is  easy  to  see,  that  by  this,  we  abridge  the  cal- 
culation considerably.  If,  therefore,  the  Swiss  make  the  assault, 
their  number  being  x,  tiiat  of  the  other  will  be  5  —  x ;  now,  the 
former  receive  1  crown,  and  the  latter  half  a  crown  ;  so  that  we 
shall  have, 

2i -f  Is  — |a;  =  901. 

We  find  in  the  same  manner,  that  if  the  Swabians  make  tlie 
assault,  we  have, 

1/  +  -IS— 1|/  =  901. 


18«  Mgchra,  Sect.  4. 

And  lastly,  that,  if  the  Saxons  mount  to  the  assault,  we  have, 

«  +  i-s  — ^25  =  901. 
Each  of  these  three  equations  will  enable  us  to  determine  one 
of  the  unknown  quantities  x,  y,  a ; 

For  the  first  gives         x  =  1802  —  s, 
the  second  gives    2y  =  2703  —  s, 
the  third  gives      5x>  =  3604  — '  s, 
If  we  now  take  the  values  of  6x,  6y,  and  6z,  and  write  those 
values  one  above  the  other,  we  shall  have, 
6x  =  10812  —  65, 
6y  =    8109  —  3s, 
625  =    7208  —  2s, 


and  adding;  6s  =  26129  —  lis,  or  17s  =  26129;  so 

that  s  =  1537  ;  this  is  the  whole  number  of  soldiers,  by  which 
means  we  find, 

a;  =  1802—  1537  =  265; 
21/  =  2703  —  1537  =  1166,  or  7/  =  583  ; 
3^  =  3604  —  1537  =  2067,  or  a  =  689. 
Answer.     The  company  of  Swiss  consists  of  265  men ;  that 
of  Swabians  583  ;  and  that  of  Saxons  689. 


CHxiPTER  Y. 

Of  the  Resolution  oj  fiire  Quadratic  Equations. 

541.  Ax  cquatimi  is  said  to  he  of  the  second  degree,  when  it  contains 
the  square  or  tJie  second  jjower  of  the  unknown  quantity ,  without  any 
of  its  higher  powers.  An  equation,  containing  likewise  the  third 
power  of  the  unknown  quantity,  belongs  to  cubic  equations,  and 
its  resolution  requires  particular  rules.  There  are,  therefore, 
only  three  kinds  of  terms  in  an  equation  of  the  second  degree. 

1.  The  terms  in  which  the  unknown  quantity  is  not  found  at 
all,  or  which  are  composed  only  of  known  numbers. 

2.  The  terms  in  which  we  find  only  the  first  power  of  tlie 
unknown  quaiiiity. 

3.  The  terms  which  contain  the  square,  or  the  second  power 
of  the  unknown  quantity. 


Chap.  5.  Cff  Compound  ^lantities*  153 

So  that  X  si^^r>ifying  an  unknown  quantity,  and  the  letters  a, 
h,  c,  d,  &c.  representing  known  numbers,  tlie  terms  of  the 
first  kind  will  have  the  form  a,  the  terms  of  the  second  kind 
will  have  the  form  bx,  and  the  terms  of  the  tiiird  kind  will  have 
the  form  cxx, 

542.  We  have  already  seen,  how  two  or  more  terms  of  the 
same  kind  may  be  united  togetlier,  and  considered  as  a  single 
term. 

For  example,  we  rtiay  consider  the  formula  axx  —  bxx  + 
cxx  as  a  single  term,  representing  it  thus  (a  —  6  +  c)  xx ; 
since,  in  fact,  (a  —  b  -j-  c)  in  a  known  quantity. 

And  also,  when  such  terms  are  found  on  both  sides  of  the 
sign  =,  we  have  seen  how  they  may  be  brought  to  one  side,  and 
then  reduced  to  a  single  term.     Let  us  take,  for  example,  the 

equation, 

2xx  —  3ar  +  4  =  5xx  —  8a?  -f  11  ; 

We  first  subtract  Qxx,  and  tliere  remains 

—  3x  +  4  =  Sxx  —  8x  4-  11  5 
then  adding  Sx,  we  obtain, 

5a?  -f  4  =  Sxx  -f  1 1  f 
Lastly,  subtracting  11,  there  remains  Sxx  =  5x  —  7. 

543.  We  may  also  bring  all  the  terms  to  one  side  of  the  sign 
=,  so  as  to  leave  only  0  on  the  other.  It  must  be  remembered, 
however,  that  when  terms  ar*c  transposed  from  one  side  to  the 
other,  their  signs  must  be  changed.^ 

Thus,  the  above  equation  will  assume  this  form,  Sxx  —  5x 
4-7  =  0;  and,  for  this  reason  also,  thefoliowing  geiieral  formula 
represents  all  equations  of  the  second  degree, 

axx  ±  bx  ±  c  =  Of 
in  wliich  the  sign  ±  is  read  plus  or  mimes,  and  indicates  that 
such  terms  may  be  sometimes  positive,  and  sometimes  negaiive. 

544.  Whatever  be  the  original  form  of  a  quadratic  equation, 
it  may  always  be  reduced  to  this  formula  of  three  terms.  If  wo 
have,  for  example,  the  equation 

ax-\-b BX'\'f 

cjc-\-d     ^x-\-li 

*  That  is,  the  quantity  thus  transposed  is  added  to  or  subtracted  frotti  each 
side  of  the  equation. 


184  dlgehra.  Sect.  4. 

we  must,  first,  reduce  the  fractions  ;  multiplying,  for  this  pur- 
pose, by  ex  +  (/,  we  have  ax-\-b=:  <^g-^'^^+cA-+e^/^4-//  ^^^^^ . 

gx  +  /i,  we  have  agxx  -f  igx  -f  ahx  +  ft/t  =  cexx  -f  c/o;  +  edx 
-f  /(/,  which  is  an  equation  of  the  second  degree,  and  reducible 
to  the  three  following  terms,  which  we  shall  transpose  by  ar- 
ranging them  in  the  usual  manner  : 

0  =  agxx  4-  hgx  -f  ft/i, 
—  cexx  '\-  ahx  — /(Z, 

—  ^foc, 

—  edx. 

We  may  exhibit  this  equation  also  in  the  following  form, 
which  is  slill  more  clear  : 

0  =  {ag  —  ce)  XX  -f  (pg  -\-  ah  —  cj —  ed)  x  +  hh  — fd, 

545.  Equations  of  tlic  second  degree,  in  which  all  the  three 
kinds  of  terms  are  found,  are  called  complete,  and  the  resolution 
of  them  is  attended  with  greater  difficulties ;  for  which  reason 
we  shall  first  consider  those,  in  which  one  of  the  terms  is  wanting. 

Now,  if  the  term  xx  were  not  found  in  the  equation,  it  would 
not  be  a  quadratic,  but  would  belong  to  those  of  which  we  have 
already  treated.  If  the  term,  which  contains  only  known  numbers, 
tvere  wanting,  the  eqnation  would  have  this  form,  axx  ±  bx  =  0, 
which  heing  divisible  by  x,  may  be  reduced  io  ax  ±  b  =  0,  which 
is  likewise  a  simple  equation,  and  belongs  not  to  the  present  class, 

546.  But  when  the  middle  term,  which  contains  the  first  power 
of  x,  is  wanting,  the  equation  assiimes  this  form,  axx  ±  c  =  0,  or 
axx  =  qp  c  ;  as  the  sign  of  c  may  be  eitiier  positive  or  negative. 

We  shall  call  such  an  equation  a  jmre  equation  of  the  second 
degree,  since  the  resolution  of  it  is  attended  with  no  difficulty  ;  for 

c 

we  have  only  to  divide  by  a,  which  gives  xx  =  —  ;  and  taking  the 

a 

square  root  of  both  sides,  we  find  x  =  ^— ;  by  means  of  which  the 

eqnation  is  resolved, 

54'.  But  there  arc  three  cases  to  be  considered  here.  In  the 
first,  when  —  is  a  square  number  (of  which  we  can  therefore 
really  assign  the  root)  7ve  obtain  for  the  value  of  x  a  rational 


Chap.  5.  Of  Compaund  Quantities,  It^ 

number  which  may  be  either  integer  or  fractmial.  For  example, 
the  equation  xx  =  144,  gives  x  =  12.     And  xx  =  ^\,  gives 

«.  ^    3 
i«^  —    4. 

The  second  variety  is,  wJien  —  is  not  a  square,  in  which  case 

d 

ive  must  therefore  be  contented  with  the  sipi  \/,  If,  for  example, 
XX  =  12,  we  have  x  =  v'Ts*  the  value  of  which  may  be  deter- 
mined by  approximation,  as  we  have  already  shewn. 

TJie  third  case  is  tlmt  in  which  —  becomes  a  negative  number ; 

then  the  value  of  xis  altogether  impossible  and  imaginary  ;  and  this 
result  proves  that  tlie  question,  which  leads  to  such  an  equation,  is 
in  itself  impossible, 

548.  We  shall  also  observe,  before  proceeding  further,  that 
whenever  it  is  required  to  extract  tlie  square  root  of  a  number, 
that  root,  as  we  have  already  remarked,  has  always  two  values,, 
the  one  positive  and  the  other  negative.  Suppose  we  Jiave  the 
equation  xx  =  49,  the  value  ofx  will  be  not  only  +  7,  but  also  —  7, 
which  is  expressed  61/  x  =  ±  7.  So  that  all  those  questions  admit 
of  a  double  answer  ;  but  it  will  be  easily  perceived  that  in  several 
cases,  in  those,  for  example,  w  hich  relate  to  a  certain  number 
of  men,  the  negative  value  cannot  exist. 

549.  In  such  equations,  also,  as  axx  =  bx,  where  the  known 
quantity  c  is  wanting,  there  may  be  two  values  of  x,  though  we 
find  only  one  if  we  divide  by  x.  In  the  equation  xx  =  Sx,  for 
example,  in  which  it  is  required  to  assign  such  a  value  of  x,  tliat 
XX  may  become  equal  to  Sx,  this  is  done  by  supposing  x  =  3, 
a  value  which  is  found  by  dividing  tlie  equation  by  x ;  but, 
beside  this  value,  there  is  also  another,  wliich  is  equally  satis- 
factory, namely  x  =  0  ;  for  then  xx  =  0,  and  Sx  =  0.  Equa- 
tions therefore  of  tlie  second  degree,  in  general,  admit  of  two  solu- 
tions, whilst  simple  equations  admit  oidy  of  one. 

We  shall  now  illustrate,  by  some  examples,  what  wc  have 
said  with  regard  to  pure  equations  of  the  second  degree. 

550.  Question  I.  Required  a  number,  the  half  of  which  mul- 
tiplied by  the  third  may  produce  24. 

Let  this  number  =  x  ;  \x,  multiplied  by  ^x,  must  give  24  ; 
w-e  shall  therefore  have  the  equation  \xx  =  24. 
24 


186  Jilgebra.  Sect  4. 

Multiplying  by  6,  we  have  xx  =  144 ;  and  the  extraction  of 
the  root  gives  x  =  ±  12.  AVe  put  ±  ;  for  if  x  =  -|-  12,  we 
have  ^x  =  6,  and  |x  =  4  :  now  the  product  of  these  two  nuin- 
hcrs  is  24 ;  and  if  x  =  —  12,  we  have  ^x  =■  —  6,  and  ^x  = — 4, 
the  product  of  which  is  likewise  24. 

551.  Q^uestiaii  II,     Required  a  number  such,  that  hy  adding 

5  to  it,  and  subtracting  5  from  it,  the  product  of  the  sum  by  th« 
difference  would  be  96. 

Let  this  number  be  x,  then  ^  -f-  5,  multiplied  by  x  —  5,  must 
give  96 ;  whence  results  the  equation,  xx  —  25  =  96. 

Adding  25,  we  have  xx  =  121 ;  and  extracting  the  root,  w« 
have  X  =.  11.     Thus  ^  -f  5  =  16,  also  x  —  5  =  6;  and  lastly, 

6  X  16  =  96. 

552.  Question  III.  Required  a  number  such,  that  by  adding 
it  to  10,  and  subtracting  it  from  10,  the  sum,  multiplied  by  the 
remainder,  or  difference,  will  give  51. 

Let  X  be  this  number  ;  10  -f  x,  multiplied  by  10  —  x,  must 
make  51,  so  that  100  —  xx  =  51.  Adding  xx,  and  subtracting 
51,  we  have  xx  =  49,  the  square  root  of  which  gives  x  =  7. 

553.  ((uestion  IV.  Three  persons,  who  had  been  playing, 
leave  off;  the  first,  with  as  many  times  7  crowns,  as  the  second 
has  three  crowns ;  and  the  second,  with  as  many  times  17 
crowns,  as  the  third  has  5  crowns.  Further,  if  we  multiply  the 
money  of  the  first  by  the  money  of  the  second,  and  the  money 
of  the  second  by  the  money  of  the  third,  and  lastly,  the  money 
of  the  third  by  that  of  the  first,  the  sum  of  these  three  products 
will  be  3830|.     How  much  money  has  each  ? 

Suppose  that  the  first  player  has  x  crowns ;  and  since  ^  e  has 
as  many  times  7  crowns,  as  tlie  second  has  3  crowns,  we  know 
that  his  money  is  to  that  of  the  second,  in  the  ratio  of  7  :  3. 

We  shall  therefore  make  7  ;  3  =  a:,  to  the  money  of  the 
second  player,  which  is  therefore  |.x. 

Further,  as  the  money  of  the  second  player  is  to  that  of  the 
third  in  the  ratio  of  17  :  5,  we  shall  say,  17  :  5  =  fr  to  the 
money  of  the  third  player,  or  to  -^VV'^* 

Multiplying  x,  or  the  money  of  the  first  player,  by  ^x?  the 
money  of  the  scccmd,  w^e  have  the  produ  ;  ^xx.  Then  ^x,  tlic 
money  of  the  second,  multiplyed  by  the  money  of  the  third,  or 


Chap.  5.  Of  Compound  Quantities,  187 

by  jYgO:,  gives  ^-^-^xx.  Lastly,  tlie  money  of  the  third,  or 
-}-Y-^x  multiplied  by  x,  or  the  money  of  the  first,  gives  -^^-^xx* 
The  sum  of  these  three  products  is  ^xx  +  //^-^x  +  jV?^^  >  ^"^> 
reducing  these  fractions  to  the  same  denominator,  we  find  their 
sum  |0|xx,  which  must  be  equal  to  the  number  3830|. 

We  have,  therefore,  |||xx  =  3830|. 

So  that  \%\^xx  =  114y2,  and  I52\xx  being  equal  to  9572836, 
dividing  by  1521,  we  havexo?  =  ^y/^V^  ?  ^"^  taking  its  root, 
we  find  X  =  ^||'*'.  This  fraction  is  reducible  to  lower  terms  if 
we  divide  by  13  ^  so  that  x  =  ^|^  =  79^  ;  and  hence  we  con- 
clude, that  ^x=  34,  and  -^^  a;  =  10. 

Answer,  The  first  player  has  79-|  crowns,  the  second  has  34 
crowns,  and  the  third  10  crowns. 

Remark,  This  calculation  may  be  performed  in  an  easier 
manner ,-  namely,  by  taking  the  factors  of  the  numbers  which 
present  themselves,  and  attending  chiefly  to  the  squares  of  those 
factors. 

It  is  evident,  that  507  =  3  x  169,  and  that  169  is  the  square 

'of  13  ;  then,  that  833  =  7  x  119,  and  119  =  7  X  17.     Now,  we 

3x169 
have    — — '-xxz=.  S830|,  and  if  we  multiply  by  3,  we  have 
17X49  ,*.... 

9x169 

— — XX  =  1H92.     Let  us  resolve  this  number  also  into  its 

17x49 

factors ;  we  readily  perceive,  that  the  first  is  4,  that  is  to  say, 

that  11492  =  4  X  2873  ;  further,  2873  is  divisible  by  17,  so  that 

2873  =  17  X  169.     Consequently,  our  equation  will  assume  the 

9x  1 69 
following  form  :  rj—j^^^  =  4  x  17  X  169,  which,  divided  by 

169,isreducedto-^:-— -  xo?  =  4  x  17  ;  multiplying  also  by  17  X 
1/  X4y 

4x289x49 
49,  and  dividing  by  9,  we  have  xx  = ,  in  which  all 

the  factors  are  squares ;  whence  we  have,  without  any  further 

calculation,  the  root  x  =   — - — -  =  -^  =  794»  as  before. 

554.  Question  V.  A  company  of  merchants  appoint  a  factor 
at  Archangel.  Each  of  them  contributes  for  the  trade,  which 
they  have  in  view,  ten  times  as  many  crowns  as  there  are  pstrt- 


Iff  Mgeh'd.  Sect.  4. 

ncrs.  The  profit  of  the  factor  is  fixed  at  twice  as  many  crowns, 
per  centt  as  there  arc  partners.  Further,  if  we  multiply  the  ^l^ 
part  of  his  total  gain  hy  2|,  the  number  of  partners  will  be 
found.     Required,  what  is  that  number  ? 

Let  it  be  =  :j;  ;  and  since,  each  partner  has  contributed  lOx, 
the  whole  capital  is  =  10a?x.  Now,  for  every  hundred  crowns, 
the  factor  gains  2a;,  so  that  with  the  capital  of  10:»a?  his  profit 
will  be  inc^.  The  ^^^  part  of  this  gain  is  -sJt^x*^  ;  multiplying 
by  2|.  or  by  y^,  we  have  4yJo  ^^'  or  g^yX^,  and  this  must  be 
equal  to  the  number  of  pai-tners,  or  x. 

We  have,  therefore,  the  equation  aij-^^  =  a:,  ora;^  =  225x; 
which  appears,  at  first,  to  be  of  the  third  degree ;  but  as  we  may 
divide  by  cc,  it  is  reduced  to  the  quadratic  xx  =  2^5,  whence 
x=z  15. 

Answer,  There  are  fifteen  partners,  and  each  contributed 
150  crowns. 


CHAPTER  VI. 

Of  the  Resolution  of  Mixt  Equations  of  the  Second  Degree, 

555.  An  equation  of  the  second  degree  is  said  to  be  mixt,  or  com- 
plete,* when  three  kinds  of  terms  are  found  in  it,  namely,  that 
which  contains  the  square  of  the  unknown  quantity,  as  axx ;  that, 
in  which  the  unknown  quantity  is  found  oidy  of  the  first  power,  as 
bx ;  lasUy,  the  kind  of  terms  which  is  composed  only  of  known 
quantities.  And  since  we  may  unite  two  or  more  terms  of  the 
same  kind  into  one,  and  bring  all  the  terms  to  one  side  of  the 
sign  =,  the  general  form  of  a  mixt  equation  of  the  second  degree 
will  be 

axx  ^:  hx  ^:.  c  z=.0. 

In  this  chapter,  we  shall  shew,  how  the  value  of  x  is  derived 
from  such  equations.  It  will  be  seen,  that  there  are  two  me- 
thods of  obtaining  it. 

55Q>,  An  equation  of  the  kind  that  we  are  now  considering, 
may  be  reduced,  by  division,  to  such  a  form,  that  the  first  term 
may  contain  only  the  square  xx  of  the  unknown  qi'aii.*it^'  x    We 

•  Sometimes  called  also  affected. 


Ghap.  6.  Of  Compound  ^lantities.  169 

sliall  leave  the  second  term  on  the  same  side  with  or,  and  trans- 
pose the  known  term  to  the  other  side  of  the  sign  =.  By  these 
means  our  equation  will  assume  the  form  xx  ±  px  =  ±  q, 
in  wliich  p  and  q  represent  any  known  numbers,  positive  or 
negative ;  and  the  whole  is  at  present  reduced  to  determining 
the  true  value  of  x.  We  shall  begin  with  remarking,  that  if  xx 
-\-p^  were  a  real  square,  the  resolution  would  be  attended  with 
no  difficulty,  because  it  would  only  be  required  to  take  the  square 
root  of  both  sides. 

557.  But  it  is  evident  that  xx  4-  px  cannot  be  a  squai'e ; 
since  we  have  already  seen,  that  if  a  root  consists  of  two  terms, 

for  example,  x  -f  n,  its  square  always  contains  three  terms, 
namelij,  twice  the  product  of  the  two  parts,  beside  the  sqiuire  of  each 
part ;  that  is  to  say,  the  square  o/"  x  -f  n  is  xx  4-  2nx  -f-  nn.  Now, 
we  have  already  on  one  side  xx  +px ;  we  may,  therefore,  con- 
sider XX  as  the  square  of  the  first  part  of  the  root,  and  in  this  case 
px  must  represent  twice  the  product  of  x,  the  first  part  of  the  root, 
by  the  second  part ;  consequently,  this  second  part  must  be  -|p,  and 
in  fact  the  square  0/*  x  +  ip,  is  found  to  be  xx  +  px  -f  -^pp. 

558.  JS^ow  XX  +  px  -f  ;^pp  being  a  real  square,  which  has  for  its 
root  X  +  ip,  if  we  resume  our  equation  xx  -f-  px  =  q,  we  have 
only  to  add  ^pp  to  both  sides,  which  gives  us  xx  +  px  -f  ipp  =  q 
4-  ipp,  the  first  side  being  actually  a  square,  and  the  other  contain- 
ing only  known  quantities.  If,  therefore,  we  take  the  square  root 
of  both  sides,  we  find  x  +  ^p  =  >/  (|  pp  +  q)  ;  and  subtracting  |  p, 
we  obtain  x  =  —  ^P  +  V  CiPP  +  q)  J  ^^^^>  os  every  square  root 
may  be  taken  either  affirmatively  or  negatively,  we  shall  have  for 
X  two  values  expressed  thus ; 

^  =  -4p*  JtPP  +  'i- 

559.  This  formula  contains  the  rule  by  which  all  quadratic 
equations  maybe  resolved,  and  it  will  be  proper  to  commit  it  to 
memory,  that  it  may  not  be  necessary  to  repeat,  every  time,  the 
whole  operation  which  we  have  gone  through.  We  may  always 
arrange  the  equation,  in  such  a  manner,  that  the  pure  square 
XX  may  be  found  on  one  side,  and  t!ie  above  equation  have  the 
form  XX  =  —  px  -\-q,  where  we  see  immediately  that  a?  =  - —  ^p 


\4 


pp  -\-q. 


w6  JUgebra.  Sect.  4. 

560.  The  general  rule,  therefore,  which  we  deduce  from  this, 
in  order  to  resolve  the  equation  xx  =  —  px  -{-  q,  ]s  founded 
on  this  consideration  ; 

That  the  unknown  quantity  x  is  equal  to  half  tlie  coefficient, 
or  multiplier  of  x  on  the  other  side  of  the  equation,  plus  or  minus 
the  square  root  of  the  square  of  this  number,  and  the  known 
quantity  which  forms  the  third  term  of  the  equation. 

Thus,  if  we  had  the  equation  xx  =  6a?  +  7,  we  should  imme- 
diately say,  that  x  =  5  ±  vsTr  =  3  ±  4,  whence  we  have 
these  two  values  of  ^,  T.  ^  =  7  ;  IL  ^  =  —  1.  In  the  same 
mariner,  the  equation  xx  =  10^  —  9,  would  ,^ive  x  =  5  ± 
V/25  — 9  =  5  ±  4,  that  is  to  say,  the  two  values  of  x  are  9  and  1. 

561.  This  rule  will  be  still  better  understood,  by  distinguish- 
ing the  following  cases.  I.  when  p  is  an  even  number ;  II. 
when  p  is  an  odd  number;  and  III.  when|)  is  a  fractional 
number. 

I.  Let  p  be  an  even  number,  and  the  equation  such,  that  xx 
=  Qpx  -f-  q ;  we  shall,  in  this  case,  have  x  =p  ±  ^pp  +  q, 

II.  Let  p  be  an  odd  number,  and  the  equation  xx  =  2)x  +  q  ; 

we  shall  here  have  x  =  ^p  ±  ^-rPP  +  9  >  ^^^  since  ipp  +  q  = 
JTJI-5,  we  may  extract  the  square  root  of  the  denominator,  and 


write  a;  =  Ap  ±  ^'""X'"  =  E^imiL. 

III.  Lastly,  if  p  be  a  fraction,  the  equation  may  be  resolved 
in  the  following  manner ;  let  the  equation  be  axx  =zbx  -\.  c,  or 

XX  =  —  +  —  9  and   we  shall   have,  by  the  rule,  x  =  —  ± 
a        a  2a 

fj^  I     ^.     Now, 1 —   = ,  the  denominator  of 

^U«a^    a  4aa^    a  4aa 

which  is  a  square  ;   so  that  x  —    L^^tiff. 

2a 

562.  The  other  method  of  resolving  mixt  quadratic  equations, 
is  to  transform  them  into  pure  equations.  This  is  done  by  sub- 
stitution ;  for  example,  in  the  equation  xx  z=z  px  -^  q,  instead  of 
the  unknown  quantity  a-,  we  may  write  another  unknown  quan- 
tity y,  such,  that  x  =z  ij  4-  Ip  ;  by  which  means,  when  we  have 
determined  y,  we  may  immediately  find  the  value  of  x^ 


Chap.  6,  Of  Compound  Qiuintities.  191 

If  we  make  this  substitution  of  1/  +  |/>  instead  of  x,  we  have 
XX  =  yy  +ptj  +  Ijj^J,  and  px  —py  -\-  ^pp ;  consequently,  our 
equation  will  become  ytj  -^  py  ^  ipp  =  py  -\-  ^pp  4.  q,  which  is 
first  reduced,  by  subtracting  py,  to  yy  +  ^pp  =  ijrp  +  q ;  and 
then,  by  subtracting  ipp,  to  yy  =  ^pp  +  ^.     This  is  a  pure 

quadratic  equation,  which  immediately  gives  y  =  ^     ]—pp  +  Q' 

Now,  since  x=y  -^  jp,we  have  x=  ^p   ±  A-^p  +  (/,   as  we 

found  it  before.  We  have  only,  therefore,  to  illustrate  this  rule 
by  some  examples. 

563»  Question  I.  There  are  two  numbers ;  one  exceeds  the 
other  by  6,  and  their  product  is  91.     Wliat  are  those  numbers  ? 

If  the  less  is  x,  the  other  is  x  -f  6,  and  their  product  xx  -|-  6x 
=  91.  Subtracting  607,  there  remains  0^07  =  91  —  6a?,  and  the 
rule  gives  x  =  —  3  ±  ^9  +  91=  —  3  ±  10;  so  that  x=7y 
and  X  =  —  13. 

Answer,     The  question  admits  of  two  solutions  ; 

By  one,  the  less  number  ;r  is  =  7,  and  the  greater  ^+6=13. 

By  the  other,  the  less  number  x  =  —  13,  and  the  greater 

564.  question  II.  To  find  a  number  such,  that  if  9  be  taken 
from  its  square,  the  remainder  may  be  a  number,  as  many  units 
greater  than  100,  as  the  number  sought  is  less  than  23. 

Let  the  number  sought  =  ^ ;  we  know,  that  xx  —  9  exceeds 
100  by  XX  —  109.  And  since  x  is  less  than  23  by  23  —  x,  we 
have  this  equation  ;  xx  —  109  =  23  —  x. 

Wherefore  xx  =  —  ^  +  132,  and,  by  the  rule,  x  =  —  |  + 

Ji  +  '32  -  -^  ±  J~  =  -  4  ±f .     So  that  ^  =  11,  and 

x  =  —  12. 

Jinsxver,  When  only  a  positive  number  is  required,  that 
number  will  be  11,  the  square  of  which  minus  9  is  112,  and 
consequently  greater  than  100  by  12,  in  the  same  manner  as  11 
is  less  tlian  23  by  12. 

565,  Question  III.  To  find  a  number  such,  that  if  we  multi- 
ply its  half  by  its  third,  and  to  the  product  add  half  the  number 
required,  the  result  will  be  30, 


192  Algebra.  Sect.  4. 

Suppose  that  number  =  ^,  its  half,  multiplied  by  its  third, 
will  make  ^07^' ;  so  that  ^xx  -f  ix  =  30.  Multiplying  by  6, 
we  have  xx  ■+■  Sx  =   180,  or  ^or  =  —  3x  +  180,  which  gives 

Consequently  x  is  either  =  12,  or  —  15. 

566.  Question  IV.  To  find  two  numbers  in  a  double  ratio  to 
each  other,  and  such  that  if  we  add  their  sum  to  their  product, 
we  may  obtain  90. 

Let  one  of  the  numbers  =  .r,  then  the  other  will  be  =  9.x ; 
their  product  also  =  ^.xx,  and  if  we  add  to  this  3x,  or  their  sum, 
the  new  sum  ought  to  make  90.  So  that  9.xx  -f  3^  =  90 ;  9.xx 
—  90  —  3^  ;  XX  z=  —  I  4-  45,  whence  we  obtain  a;  =  —  |  + 


4 


9    ,    ^r.  3      27 

—   4-45    = +— . 

16  ^  4—4 


Consequently  x  =  6,  or  —  7^-. 

567.  Question  V.  A  horse  dealer,  who  bought  a  horse  for  a 
certain  number  of  crowns,  scels  it  again  for  119  crowns,  and  his 
profit  is  as  much  per  cent,  as  the  horse  cost  him.  Required, 
what  he  gave  for  it  ? 

Suppose  tlie  horse  cost  x  crowns  ;  then  as  the  horse  dealer 
gains  a- per  cent   we  shall  say,  if  100  give  the  profit  ;r,*  what 

does  X  ffive  ?    Answer,  ^-^,  Since  therefore  he  has  gained  - — , 
*  100  ^  100' 

and  the  horse  originally  cost  him  x  crowns,  he  must  have  sold 
it  for  X  +  — — ;  wiierefore  x  ^ =119.     Subtracting  x, 

we  have  — —  =  —  x  -{-  119  ;  and  multiplying  by  100,  we  have 

XX  =  —  lOOx  -f  11900.     Applying  the  rule,  we  find  x  =  — 
50  +  v/i^500  +  11900  =  —  50  +  VUioo"  =  —  50  ±   120. 

Answer,  The  horse  cost  70  crowns,  and  since  the  horse 
dealer  gained  70  per  cent,  when  he  sold  it  again,  the  profit  must 
have  been  49  crowns.  The  horse  must  have  been  therefore 
sold  again  for  70  +49,  that  is  to  say  for  119  crowns. 

568.  Question  VI.  A  person  buys  a  certain  number  of  pieces 
of  clotli :  he  pays,  for  the  first,  2  crowns ;  for  the  second,  4 
crowns  ;  for  the  third,  6  crowns,  and  in  the  same  manner  always 


Chap.  6.  Of  Compound  Quantities.  193 

2  crowns  more  for  each  following  piece.  Now,  all  the  pieces 
together  cost  him  110.     How  many  pieces  had  he  ? 

Let  the  number  sought  =  x.     By  the  question,  the  purchaser 
paid  for  the  different  pieces  of  cloth  in  the  following  manner ; 
for  the     1,  2,  3,  4,  5  ....  a; 
he  pays    2,  4,  6,  8,  10  ....  20?  crowns. 

It  is  therefore  required  to  find  tlie  sum  of  the  arithmetical 

progression  24-4-f-6-f-8-fl0-f 2x,  which  consists  of 

X  terms,  that  we  may  deduce  from  it  the  price  of  all  the  pieces 
of  cloth  taken  together.  The  rule  which  we  have  already  given 
for  this  operation,  requires  us  to  add  the  last  term  and  the  first; 
the  sum  of  which  is  20?  -f  2  ;  if  we  multiply  this  sum  by  the 
number  of  terms  x,  the  product  will  be  9.xx  -j-  2a; ;  if  we  lastly 
divide  by  tlie  difference  2,  the  quotient  will  be  xx  -f-  x,  which 
is  the  sum  of  the  progression  ;  so  that  we  have  xx  -f  a?  =  110  ; 

wherefore  xx  =  —  x  -{-  1 10,  and  x= --f-\  —  +110=  — 

i  +  i^  =  10. 

2         2. 

Answer.     The  number  of  pieces  of  cloth  is  10. 

569.  Question  VII.  A  person  bought  several  pieces  of  cloth, 
for  180  crowns.  If  he  had  received  for  the  same  sum  3  pieces 
more,  he  would  have  paid  three  crowns  less  for  each  piece  ? 
How  many  pieces  did  he  buy  ? 

Let  us  make  the  number  sought  =  x ;  tlicn  each  piece  will 

1  OQ 

have  cost  him  - —  crowns.  Now,  if  the  purchaser  had  had  a;  -f  3 

1 80 

pieces  for  180  crowns,  eacli  piece  would  have  cost crowns  : 

and,  since  this  price  is  less  than  the  real  price  by  three  crowns, 
we  have  this  equation, 

180    _180         ^ 


Multiplying  by  x,  w^e  have  — ~  =180  —  3.^;  dividing  by 

60a? 
3,  we  have  -  =  60  —  x;  multiplying  by  a?  +  3  we  have 

60o:  =  180  +  57 X  —  XX ;  adding  xx,  wc  shall  have  xx  +  60x 

25 


194  M^snebrw,  Sect.  4. 


"S 


=  180  +  57 X*,  subtracting  60a^,  we  shall  have  xx:=:.  —  Soj  + 
180. 
The  rule,  consequently,  gives 

a-  =  —  I  +  Vf  +  180,  or  o:  =  —  I  +^  =  12. 

Answer,  He  bought  for  180  crowns  12  pieces  of  cloth  at  15 
crowns  the  piece,  and  if  he  had  got  3  pieces  more,  namely  15 
pieces  for  180  crowns,  each  piece  would  have  cost  only  12 
crowns,  that  is  to  say,  3  crowns  less. 

570.  Question  VIII.  Two  merchants  enter  into  partnership 
with  a  stock  of  100  crowns ;  one  leaves  his  money  in  the  part- 
nership for  three  months,  the  other  leaves  his  for  two  months, 
and  each  takes  out  99  crowns  of  capital  and  profit.  What  pro- 
portion of  the  stock  did  each  furnish  ? 

Suppose  the  first  partner  contributed  x  crowns,  the  other  will 
have  contributed  100  —  x.  Now,  the  former  receiving  99 
crowns,  his  profit  is  99  —  x,  which  he  has  gained  in  three 
months  with  the  principal  x ;  and  since  the  second  receives  also 
99  crowns,  his  profit  is  a;  —  1,  wliich  he  has  gained  in  two 
months  with  the  principal  100  —  x  ;  it  is  evident  also,  that  the 

profit  of  this  second  partner  would  have  been    '  ""*  ,  if  he  had 

remained  three  months  in  the  partnership.     Now,  as  the  profits 

gained  in  the  same  time  are  in  proportion  to  the  principals,  we 

3^—3 
have  the  following  proportion,  a?  :  99  —  x=  100  —  x  :   — - — . 

The  equality  of  the  product  of  the  extremes  to  that  of  the 
means,  gives  the  equation, 
5.T.V — S^ 


9900  —  199.T  -f  XX  ; 


2 

Multiplying  by  2,  we  have  5xx  —  3a:  =  19800  —  398x 
4-  Qxx  ;  subtracting  2xXf  we  have  xx  —  3x  =  19800  —  398a; 
adding  5x,  we  have  xx  =  19800  —  395a:. 
Wherefore,  by  the  rule, 

S95     ,    Jl56025    ,    79^  _  __  395        485  ^90^^^ 

Answer,    The  first  partner  contributed  45  crowns,  and  the 
other  55  crowns.    The  first,  having  gained  54  crowns  in  three 


Chap.  6.  Of  Compound  ^uintities,  195 

months,  would  have  gained  in  one  month  18  crowns ;  and  the 
second  having  gained  44  crowns  in  two  months,  would  have 
gained  22  crowns  in  one  month  :  now  these  profits  agi'ee ;  for, 
if  with  45  crowns  18  crowns  are  gained  in  one  month,  22 
crowns  will  be  gained  in  t!ie  same  time  with  55  crowns. 

571.  (fiiestion  IX.  Two  girls  carry  100  eggs  to  market ;  one 
had  more  than  the  other,  and  yet  the  sum  which  they  hoth 
received  for  them  was  the  same.  Tlie  first  says  to  the  second, 
if  I  had  had  your  eggs,  I  should  have  received  1 5  sous.  The 
other  answers,  if  I  had  had  yours,  I  should  have  received  6|. 
sous.     How  many  eggs  did  each  carry  to  market  ? 

Suppose  the  first  had  x  eggs  ;  then  the  second  must  have  had 
100  —  X. 

Since  therefore  the  former  would  have  sold  100  —  x  eggs  for 

15  sous,  we  have  the  following  proportion  ; 

i5x 
100  —  X  :  15  =  X ....  to  sous. 

100 — jc 

Also,  since  the  second  would  have  sold  x  eggs  for  6|  sous,  we 

find  how  much  she  got  for  100  —  x  eggs,  by  saying 

20       ,^^  ^     2000—2007 

X  :  —  =  100  —  X  ....  to  . 

3  Sx 

Now  both  the  girls  received  the  same  money ;  we  have  con- 

15o7  2000 20  JC 

sequently  the  equation,  — — — -   = ,  which  becomes 

this, 

25xx  =  200000  —  4000X  ; 
and  lastly  this, 

XX  =  —  160a:  +  8000  ; 
whence  we  obtain 

X  =  —  80  +  V6400  +  80U0  =  —  80  +  120  =  40. 
Answer,     The  first  girl  had  40  eggs,  the  second  had  60,  and 
each  received  10  sous. 

572.  ^uestmn  X.  Two  merchants  sell  each  a  cei'tain  quantity 
of  stuff;  the  second  sells  Sells  more  than  the  first,  and  they 
received  together  35  crowns.  The  first  says  to  the  second,  I 
should  have  got  24  crowns  for  your  stuff;  the  other  answers, 
and  I  should  have  got  for  yours  12  crowns  and  a  half.  How 
many  ells  had  each  ? 

Suppose  the  first  had  x  ells  ;  then  the  second  must  have  had 


195^                                     Mgebra,  Sect.  4. 

a:  +  S  ells.     Now,  since  the  first  would  have  sold  a:  +  3  ells  for 
^4  crowns,  he  must  have  received crowns  for  his  x  ells. 

JT  +  S 

And  with  regard  to  the  second,  since  he  would  have  sold  x  ells 
for  12-1  crowns,  he  must  have  sold  his  a:  4-  3  ells  for     ^'^"'"^ ; 

so  that  the  whole  sum  they  received  was f-  — — —  =35 

crowns. 

This  equation  becomes  xx  =  QOx  —  75,  whence  we  have  x 


=  10  ±  VIOO—  75  =  10  ±  5. 

Answer.  The  question  admits  of  two  solutions  :  according  to 
the  first,  the  first  merchant  had  15  ells,  and  the  second  had  18  ; 
and  since  the  former  would  have  sold  18  ells  for  24  crowns,  he 
must  have  sold  his  15  ells  for  20  crowns  ;  the  second,  who  would 
have  sold  15  ells  for  12  crowns  and  a  half,  must  have  sold  his 
18  ells  for  15  crowns;  so  that  they  actually  received  35  crowns 
for  their  commodity. 

According  to  the  second  solution,  the  first  merchant  had  5 
ells,  and  the  other  8  ells ;  so  that,  since  the  first  would  have 
sold  8  ells  for  24  crowns,  he  must  have  received  ]  5  crowns  for 
his  5  ells ;  and  since  the  second  would  have  sold  5  ells  for  12 
crowns  and  a  half,  his  8  ells  must  have  produced  him  20  crowns. 
The  sum  is,  as  before,  35  crowns. 


CHAPTER  VII. 

Of  the  J^ature  of  Equations  of  the  Second  Degree, 

573.  "What  we  have  already  said  sufficiently  shews,  that 
equations  of  the  second  degree  admit  of  two  solutions  ;  and  this 
property  ouglit  to  be  examined  in  every  point  of  view,  because 
the  nature  of  equations  of  a  higher  degree  will  be  very  much 
illustrated  by  such  an  examination.  We  shall  therefore  retrace, 
with  more  attention,  the  reasons  which  render  an  equatiqn  of 
tlie  second  degree  capable  of  a  double  solution  ;  since  they  un- 
doubtedly will  exhibit  an  essential  property  of  those  equations. 


Chap.  7,  Of  Compound  Quantities,  t^ 

574.  We  have  already  seen,  it  is  true,  that  this  double  solu- 
tion arises  from  the  circumstance  that  the  square  root  of  any 
number  may  be  taiven  either  positively,  or  negatively  ;  however, 
as  this  principle  will  not  easily  apply  to  equations  of  higher 
degrees,  it  may  be  proper  to  illusti-ate  it  by  a  distinct  analysis. 
Taking,  for  an  example,  the  quadratic  equation,  xx  =  12x  —  35, 
we  shall  give  a  new  reason  for  this  equation  being  rcsolvible  in 
two  ways,  by  admitting  for  x  the  values  5  and  7,  both  of  which 
satisfy  the  terms  of  the  equation. 

575.  For  this  purpose  it  is  most  convenient  to  begin  with 
transposing  the  terms  of  the  equation,  so  that  one  of  the  sides 
may  become  0 ;  this  equation  consequently  takes  the  form  xx 
—  IQx  -f-  35  =  0  ;  and  it  is  now  required  to  find  a  number 
such,  that,  if  we  substitute  it  for  x,  the  quantity  xx  —  12x  +  35 
maybe  really  equal  to  nothing;  after  tliis,  we  shall  have  to 
shew  how  this  may  be  done  in  two  ways. 

576.  Now,  the  whole  of  this  consists  in  shewing  clearly,  that 
a  quttntity  of  the  form  xx  —  12x  +  35  may  be  considered  as  the 
product  of  two  factors  ;  thus,  in  fact,  the  quantity  of  which  we 
speak  is  composed  of  the  two  factors  (x  —  5)  x  (x  —  7).  For, 
since  this  quantity  must  become  0,  we  must  also  have  the  pro- 
duct {x  —  5)  X  (.tr  —  7)  =  0  ;  but  fl  product,  of  whatever  7i«m- 
ber  of  factors  it  is  composed,  becomes  =  0,  only  when  one  of  those, 
factors  is  reduced  to  0  ;  this  is  a  fundamental  principle  to  which 

we  must  pay  particular  attention,  especially  when  equations  of 
several  degrees  ar'C  treated  of. 

577.  It  is  therefore  easily  understood,  tliat  the  product  (x  —  5) 
X  (x  —  7)  may  become  0  in  two  ways  :  one,  wlien  the  first  factor 
X  —  5  =  0;  the  other,  wJien  the  second  factor  x  —  7=0.  In  the 
first  case  x  =  5,  in  the  other  x  =  7.     TJje  reason  is  therefore 

very  evident,  why  such  an  equation|3?x  —  12x  +  35  =  0,  ad- 
mits of  two  solutions ;  that  is  to  say,  why  we  can  assign  two 
values  of  x,  both  of  which  equally  satisfy  the  terms  of  the  equa- 
tion. This  fundamental  principle  consists  in  this,  that  the 
quantity  xx  —  IQx  -f-  35  may  be  represented  by  the  product  of 
two  factors. 

578.  The  same  circumstances  are  found  in  all  equations  of 
the  second  degree.     For,  after  having  brought  all  the  terms  to 


198  Mgebra,  Sect.  4. 

one  side,  we  always  find  an  equation  of  the  following  form  ccx 
—  (WO  -j-b  =z  0,  and  this  formula  may  be  always  considered  as 
the  product  of  two  factors,  which  we  shall  represent  by  {x  —  p) 
X  (x  —  q)f  without  concerning  ourselves  what  numbers  the 
letters  p  and  q  represent.  Now,  as  this  product  must  be  =  0, 
from  the  nature  of  our  equation  it  is  evident  that  this  may  hap- 
pen in  two  ways  ;  in  the  first  place,  when  x  =  p ;  and  in  the 
second  place,  when  x  =  q;  and  these  are  the  two  values  of  x 
which  satisfy  the  terms  of  the  equation. 

579.  Let  us  now  consider  the  nature  of  these  two  factors,  in 
order  that  the  multiplication  of  the  one  by  the  other  may  exactly 
produce  xx  —  ax  -{-  b.  By  actually  multiplying  them,  we  get 
XX  —  (^P  -h  Q)  ^  -h  pq  y  now  this  quantity  must  be  the  same  as 
XX  —  ax  -\-bf  wherefore  we  have  evidently  p  -\.  q  =  a,  and  pq 
=  b.  So  that  we  have  deduced  this  very  remarkable  property, 
that  in  every  equation  of  the  form  xx  —  ax  +  b  =0,  the  two 
values  of  x  are  such,  that  their  sum  is  equal  to  a,  and  their  prodiict 
equal  to  b  ;  whence  it  follows  that,  if  we  know  one  of  the  values, 
the  other  also  is  easily  found, 

580.  "We  have  considered  the  case  in  which  the  two  values 
of  X  are  positive,  and  which  requires  the  second  term  of  the 
equation  to  have  the  sign  — ,  and  the  third  term  to  have  the 
sign  -f .  Let  us  also  consider  the  cases  in  whicli  either  one,  or 
both  values  of  x  become  negative.  The  first  takes  place,  when 
the  two  factors  of  the  equation  give  a  product  of  this  form 
(x  —  Ji)  X  (pc  -^  q)  ;  for  then  the  two  values  of  x  are  x  =  p, 
and  X  =  —  q;  the  equation  itself  becomes  xx  -f  (</  — p)  x  — 
pqz=0  ;  the  second  term  has  the  sign  -f ,  w  hen  q  is  greater  than 
p,  and  the  sign  — ,  when  q  is  less  than  p ;  lastly,  the  third  term 
is  always  negative. 

The  second  case,  in  which  both  values  of  x  are  negative, 
occurs,  when  the  two  factors  are  (x  +  p)  x  (^  +  q)  ;  for  w^e 
shall  then  have  x  =  —  p  and  x  =  —  q  ;  the  equation  itself  be- 
comes xx  -f.  (p  -f-  (/)  X  -f  2?(/  =  0,  in  which  both  the  second  and 
third  terms  are  affected  by  the  sign  -f . 

581.  The  signs  of  the  second  and  tlie  third  term  consequently 
shew  us  tlie  nature  of  the  roots  of  any  equation  of  the  second 
degree.     Let  the  equation  be  xx  . . . .  aa^ . , . .  &  =  0,  if  the 


Chap.  7.  Cff  Compound  ^lantities,  199 

second  and  third  terms  have  the  sign  +,  the  two  values  of  x  are 
both  negative  ;  it'  the  second  term  has  the  sign  — ,  and  tiie  third 
term  has  -f ,  both  values  are  positive ;  lastly,  if  the  third  term 
also  has  the  sign  — ,  one  of  the  values  in  question  is  positive. 
But  in  all  cases,  whatever,  the  second  term  contains  the  sum 
of  the  two  values,  and  the  third  term  contains  their  product. 

582.  After  what  has  been  said,  it  will  be  very  easy  to  form 
equations  of  the  second  degree  containing  any  two  given  values. 
Let  there  be  required,  for  example,  an  equation  such,  that  one 
of  the  values  of  jc  may  be  7,  and  the  other  —  3.  We  first  form 
the  simple  equations  jc  =  7  and  x  =  —  3  ;  thence  these,  x  —  7 
=  0  atid  :r  +  3  =  0,  which  give  us,  in  this  manner,  the  factors  of 
the  equation  required,  which  consequently  becomes  xx  —  4^  — 
21  =  0.  Applying  here,  also,  the  above  rule,  we  find  the  two 
given  values  of  x ;'  for  if  xx  =  Ax  ^  21,  we  have  x  = '2.  ±  \/25 
=  2  ±  5,  that  is  to  say,  x  =  7i  ov  x  =  —  S. 

583.  The  values  of  x  may  also  happen  to  be  equal.  Let  there 
be  sought,  for  example,  an  equation,  in  which  both  values  may 
be  =  5.  The  two  factors  will  be  {x  —  5)  x  (^  —  5),  and  the 
equation  sought  will  be  :r:c  —  10^  -f-  25  =  0.  In  this  equation, 
X  appears  to  have  only  one  value  ;  but  it  is  because  x  is  twice 
found  =  5,  as  the  common  method  of  resolution  shew^s  ;  for  we 
have  XX  z=lQx  —  25  ;  wherefore  x  =  5  ±  v^o"  =  5  ±  0,  that 
is  to  say,  x  is  in  two  ways  =  5. 

584.  A  very  remarkable  case,  in  which  both  values  of  x  be- 
come imaginary,  or  impossible,  sometimes  occurs ;  and  it  is 
then  wholly  impossible  to  assign  any  value  for  07,  that  would 
satisfy  the  terms  of  the  equation.  Let  it  be  proposed,  for  ex- 
ample, to  divide  the  number  10  into  two  paiis,  such,  that  their 
product  may  be  SO.  If  we  call  one  of  those  parts  x,  the  other 
will  be  =  10  —  X,  and  their  product  will  be  10  o?  —  xx  =  SO ; 
wherefore  xx-=.  \0x  —  SO,  and  x  =  5  ±  v/^^  which  being 
an  imaginary  nnmber,  shews  that  the  question  is  impossible, 

585.  It  is  very  important,  therefore,  to  discover  some  sign, 
by  means  of  which  we  may  immediately  know,  whether  an 
^nation  of  the  second  degree  is  possible,  or  not. 

Let  us  resume  the  general  equation  ax  —  xx  -f-  ft  =  0, 


200  Mgeh'u.  Sect.  4* 

>\^e  shall  have  xx  ==  ax  —  b,  and  x  =  —a±\JLaa  —  b. 

This  shews,  that  if  b  is  greater  than  Jaa,  or  4b  greater  than  aa^ 
the  two  values  of  x  are  always  imaginary,  since  it  would  be 
requii-ed  to  extract  the  scjuare  root  of  a  negative  quantity  ;  on 
the  contrary,  if  b  is  less  than  ^aa,  or  even  less  than  0,  that  is  to 
say,  is  a  negative  number,  both  values  will  be  possible  or  real. 
But,  whether  the^  be  real  or  imaginary,  it  is  no  less  true,  that 
they  are  still  expressible,  and  always  have  this  property,  that 
their  sum  is  =  a,  and  their  product  =  6.  In  the  equation  xx 
—  6^7  4-  1 0  =  Oj  for  example,  the  sum  of  the  two  values  of  or  must 
be  =  6,  and  the  product  of  these  two  values  must  be  =  10  j  now 
we  find,  I.  a?  =  3  +  v^T]  and  11,  x  =  3  —  V^^  quantities 
whose  sum  =  6,  and  the  product  =10. 

586.  The  expression  which  we  have  just  found,  may  be 
represented  in  a  manner  more  general,  and  so  as  to  be  applied 
to  equations  of  this  fovmffxx  ±  gx  -{-  h  =  0 ;  for  this  equation 

^ives  ^07  =  ±  ^^  —  4  andar=  ±  4.  ±  V-^  —  -'  «r 
oo  =  — j]_v  SF—^A.  whence  we  conclude,  that  the  two  values 

are  imaginary,  and  consequently  the  equation  impossible,  when 
4fh  is  greater  than  gg ;  that  is  to  say,  when,  in  the  equation 
fxx  —  g  a7-f  h  =z  Of  four  times  the  product  of  the  first  and  the 
last  term  exceeds  the  square  of  the  second  term  :  for  tiie  product 
of  the  first  and  the  last  term,  taken  four  times,  is  4fhxXf  and 
the  square  of  the  middle  term  is  ggxx ;  now,  if  Ajhxx  is  greater 
than  ggxx,  4fh  is  also  greater  than  gg,  and,  in  that  case,  the 
equation  is  evidently  impossible.  In  all  other  cases  the  equa- 
tion is  possible,  and  two  real  values  of  x  may  be  assigned. 
It  is  true,  they  are  often  irrational ;  but  we  have  already  seen, 
that,  in  such  cases,  we  may  always  find  them  by  approxima- 
tion,* whereas  no  approximations  can  take  place  with  regard  to 
imaginary  expressions,  such  as  \/^^;  for  100  is  as  far  from 
being  the  value  of  that  root,  as  1,  or  any  other  number. 

58r.  AYe  have  further  to  observe,  that  any. quantity  of  the 

secoTul  degree,  xx  ±  ax  ±  b,  miist  always  be  resolvible  into  two 

factors,  such  as  (a;  ±  p)  x  (pc  ±  q).    For,  if  we  took  three 


Chap.  r.  Of  Compound  ^lantities.  201 

factors,  such  as  these,  we  should  come  to  a  quantity  of  the  third 
degree,  and  taking  only  one  such  factor,  we  should  not  exceed 
the  first  degree. 

It  is  therefore  certain  that  every  equation  of  the  second  degree 
necessarily  ccndains  two  values  of  x,  and  that  it  can  neither  have 
more  nor  less, 

588.  We  have  already  seen,  that  when  the  two  factors  are 
found,  the  two  values  of  x  are  also  known,,  since  each  factor 
gives  one  of  those  values,  when  it  is  supposed  to  he  =  0.  The 
converse  also  is  true,  viz.  that  when  we  have  found  one  value  of 
jCf  we  know  also  one  of  the  factors  of  the  equation  ;  for  if  a:  =  p 
represents  one  of  the  values  of  j^,  in  any  equation  of  the  second 
degree,  x  —  p  \s  one  of  the  factors  of  that  equation  ;  that  is  to 
say,  all  the  terms  having  been  brought  to  one  side,  the  equation 
is  divisible  by  x  —  p  ;  and  further,  the  quotient  expresses  the 
other  factor. 

589.  In  order  to  illustrate  what  we  have  now  said,  let  there 
be  given  the  equation  .tjt  -f  4.t  —  21  =  0,  in  which  we  know 
that  J?  =  3  is  one  of  the  values  of  :c,  because  3x3^  +  4  x  3^ 
—  21  =  0 ;  this  shews,  that  x  —  3  is  one  of  the  factors  of  the 
equation,  or  that  xx  -\.  Ax  —  21  is  divisible  by  ^  —  3,  wliich 
the  actual  division  proves. 

a:  —  3)  XX  -f-  Ax  —  21  (x  +  7 

t/C*Ay       '  '    '     Off/ 


7x  —  21 
7a:—  21 

0. 
So  that  the  other  factor  is  x  -f  7,  and  our  equation  is  repre- 
sented by  the  product  (x  —  3)  x  (a;  -f-  7)  =  0  ;  whence  the  two 
values  of  x  immediately  follow,  the  fii^t  factor  giving  x  =  3. 
and  the  other  x  =.  —  7. 


26 


QUESTIONS  FOR  PRACTICE. 

Fractions, 

3ECTI0X  I.     CHAPTER  8. 

1.  Reduce      .  ,    _ —  to  its  lowest  terms.     Ans,  — . 

2.  Reduce    ,  ."TT     .\  to  its  lowest  terms.    Ans,  f^ , 

3.  Reduce  "^^ — -  to  its  lowest  terms.   Ans,  ^     — . 

x^ — h^x^  x^ 

4.  Reduce  — - — ^  to  its  lowest  terms.   Ans, . 

5.  Reduce  -:; — -— — -  to  its  lowest  terms.  Ans,  -—^  . 

a^ — a*a- — ax^-\-x^  1 

6.  Reduce  —, ! — I ^ to  its  lowest  terms. 

a^x-\-2a^x^-\-2ax^'\-x'^ 

Ans,  — ■ — 3. 

a^x-\-ax^-\-x 


SECTION  I.     CHAPTER  9. 

2r  b 

7.  Reduce  -^—  and  —  to  a  common  denominator. 

a  c 

^      9cx       J  ab 

Ans. and  — . 

ac  ac 

8.  Reduce  -?  and  -i-  to  a  common  denominator. 

b  c 

^       ac       ,  ah-\-h* 
Ans,  i—  and  — - —  . 
be  be 

^x     Qb 

9.  Reduce ,  — ,  and  d  to  fractions  having  a  common  de- 

2a      3c  ^ 

.     ,  ^       9cx    4ab       J  6acd 

nominator.  Ans,  - — ,  - —  ana  - — . 

oac     bac  bac 


^lestians  for  Practice.  20S 

10.  Reduce  — ,  — ^  and  a  +  ^  to  a  common  denominator. 

4     3  a 


9a     Sax         ,    1 -.a  »  4-24^7 

Ans. , ,  and , 

12a     12a  i'^a 


11.  Reduce  --,  — ,  and  ^       — >  to  a  common  denominator. 

2    3  oe+a 

3x-{-5a    2a2jH-2rt3    6.T»-f^g* 
6a:4-6a'      ixv-\-ba   '     6ji"+(ia 

12.  Reduce ,  — ,  and  — ,  to  a  common  denominator. 

2a2  '  2a  a 


2a^h   2a^c         ,  4a^d 
Alls.  — 7 ,  —-■  ,  and  — - , 
4a*     4a*'  4a* 


SECTION  I,     CHAPTER  10. 

13.  Required  the  product  of  ^  and  ^  .  Ans,   --. 

rVJ  4^  1  0 1?  4'IC  "^ 

14.  Required  the  product  of-,  -4-9  and  --^.  Aas.  -1—. 

15.  Required  the  product  of  —  and .  Ans.  —-^ — . 

a  a-\-c  a^-[-ac 

16.  Required  the  product  of  —  and  -r-.  Ans.  ^—-. 

2x         3  v*  3t^ 

IT.  Required  the  product  of  —  and  — — .   Ans,  -4 — . 

18.  Required  the  product  of  —  , ,  and  — — .     Ans,  9ax. 

^  '^  a       c  9.b 

19.  Required  the  product  of  6  4-  —  and  — .    Ans.     '^  *^ 

ax  X 

20.  Requii'ed  the  product  of  — r and  " 


be  b-{-c 


Ans. 


b^c-\-bc^  ' 


21.  Required  the  product  of  x,  ^"^  -,  and 


a:— .1 

a^-{-ab 


22.  Required  the  quotient  of  ~  divided  by  ^.    Ans.  ll. 


204  Algebm. 

23.  Required  the  quotient  of^  divided  by  Ij.  Jins.  — . 

b  d  2bc 

24.  Required  the  quotient  of   "^       ,   divided  by  ^^       . 

Ans. -^- — rrs— . 

25.  Required  the  quotient  of  «— divided  by  - 


Ans. 


•4-0. 
Ox 


x^ — ax -{-a* 

26.  Required  the  quotient  of-^'  divided  by  -.   Ans.  — . 

^  5  •'is  60 

27.  Required  the  quotient  of divided  by  5x.    Ans.  — . 

28.  Required  the  quotient  of  ^^  divided  by  ^-.    Ans.  ^-^t} . 

6  3  4a: 

29.  Required  the  quotient  of  — _f  divided  by  ^.  Ans.  :^^. 

SO.  Required  the  quotient  of —7^^  ,,  divided  by  f!±^. 

b* 
Ans.  x-i — . 

X 

Infinite  Series. 

SECTION  II.     CHAPTER  5. 

31.  Resolve  — '—   into  an  infinite  series. 

a— a? 

x^  x^         x^ 

Ans.  X  H — —  +  -^^  -f-  -^^y  kc. 

32.  Resolve  — --    into  an  infinite  series. 

a-i-x 


f-i        X   ^    X*  -^"^    .     •     \ 


0% 
33.  Resolve  — ,  into  an  infinite  series, 
or  4-6 

'34.  Resolve  -— —  into  an  infinite  series. 
1 — X 

Ans,  1  4-  %x  +  2^*  +  2^3  ^  2^4^  ^^.^ 


^estioiis  for  Practice,  205 


j8 

35,  Resolve  -; rs-  into  an  infinite  series 


Ans,  1 ^ — ,  &c. 


Surds  or  Irrational  Miinbers, 

SECTION   II.     CHAPTER   8,    &C. 

36.  Reduce  6  to  the  form  of  v^T     ^ns.  v/36. 

37.  Reduce  a  -f  6  to  the  form  of  v^Ac.     *^fis.  \/  (aa  +2a6  +  bV) 

38.  Reduce  ,  ^—  to  the  form  of  wT.     •ins,  \f -f^. 

0\/c  ^    bbc 

3 

39.  Reduce  o^  and  ft    to  the  common  index  — . 

o 

6]l  9]1 

^iw.  a  p,  and  6^p. 

40.  Reduce  v'is  to  its  simplest  form.     Am,  4  v'sT 

41.  Reduce  \/{a^x — a^x^)  to  its  simplest  form. 

^^15.  a  V(«x  —  xxy 


42.  Reduce  A  '?^    to  its  simplest  form. 


a      Sab 
Ms, 


-  lJZ 

43.  Add  v/6"  to  2  x^e]  and  vF  to  y^so.    •^'w.  3^/6";  and  V98. 

44.  Add  \/4a  and  v^IIe  together.     ,ins,  (a  +  2)  v^^ 

6ft-f-cc 


45.  Add  - 

c 


2  and 


together.    ,^iis. 


bs/bc 

4 

46.  Subtract  \/4a  fmm  v^as .     .^^i<s.  (a  —  2)  \/a. 

47.  Subtract  4*  ^om  -^.     Jns.  ^.tzS£  Jl. 

61  c  b       ^  be 

48.  Multiply  \jl±hy\j^,    Ans,^~, 

oc                  2o                       C 
3  3  

49.  Multiply  i/d  by  ^ab»     Jins,  ^(^a^b^d^). 


206  Algebra. 


50.  Multiply  'v/(4a —  Sx)  by  2a.     Ms.  \/l(>a^^  i-M^x), 

51.  Multiply  ^  \/(^^=r^)  by  (c  —  d)  v^. 


-      ac — fltt 

Alls.  —^  ^{a^x^ajc^). 

52.  Multiply  \/{^  VW^  \/3))  by  V(I7  \/(T^  v/3)). 

*  i  —  i  _*  Wl— -w 

53.  Divide  a^  by  a*  :  and  a"  by  a'\    Ans.  a^^  and  a  . 

54.  Divide  ?^^  VCa^^  — «^*)  by  ^  V("-^). 


J.  (c  —  d)  \/^x. 

55.  Divide  a^  -^  ad  —  b -^  d  \/T  by  a  —  y/dT 

Ans.  a  +  \/b  —  d* 

56.  What  is  the  cube  of  v^Il     .^ns.  ^/sT 

8  3 

57.  What  is  the  square  3  v/^cil     Ans.  9c  \/b*V. 

58.  What  is  the  fourth  power  of  |^  \f-^  ? 


.^ws. 


464(c2— 26c+62) 

59.  What  is  the  square  of  3  +  ^5*  ?     -^ns.  14  +  6  v^T 

j[  _ 

60.  What  is  the  square  root  of  a^  ?     Ans.  a^  ;  or  y/as. 

1  3   

61.  What  is  the  cube  root  of  afe^  ?    Jins.  ahU^  ;    or  \/abb. 


62.  What  is  the  cube  root  of  \/{a^  —  a:^)  ?    Ans.  \/{a^  —  x^). 

63.  What  is  the  cube  root  of  a^  —  \^{ax  —  x^)  ? 


3 

Ans. 


^(^ax  —  \/(ax  '—x*)), 

64.  What  multiplier  will  render  a  +  ys"  rational  ? 

Ans.  a  —  v^sT 

65.  What  multiplier  will  render  ^/a—  v'6~  rational  ? 

66.  What  multiplier  will  render  the  denominator  of  the  frac- 
tion — :=^-5 — —  rational  ?    Ans.  wr^  v/sT 

V7  +  V3      

67.  Resolve  \a^  -f  x^  into  an  infinite  series. 

x^  x^     .     a:6  5Jf^      j.^ 

Jin.,  a  +  _-_+_-  .^^^,  &c. 


^estions  for  Practice,  tOZ 


68.  Resolve  Vl  +  1  into  an  infinite  series. 


^„..l+^_i+_l__L^,&c. 


69.  Resolve  \a^  —  x^  into  an  infinite  series. 


Ans.  a  —  ^ — -r — -,  &c. 


a  

70.  Resolve  \  1  —  x^  into  an  infinite  series. 


^n*.l--    --    -_,&c. 

71.  Resolve  V***  —  ^*  i"*^  ^"  infinite  series. 

-  j;2  X*  x^  5x8 

72.  Resolve  ■•  into  an  infinite  series. 

73.  Resolve  (a^  —  ^*)^  into  an  infinite  series. 

\        ,,  x^  1x^  6ar6 

^ns.  aT  X  (1  -^-^  -  — ,  -  j^.   -  &c. 

74.  Resolve     ["^  "^  ^'  into  an  infinite  series. 


a«  —  J?* 


^2  ir»4  <y«6 


3 


75.   Resolve     f  ^^^  +  -^^     into  an  infinite  series. 

V(a2  J^x^Y 

1  ,  2^:2        5j:*  40x« 


Summation  of  Arithmetical  Progressions. 

SECTION  III.      CHAPTER  4. 

76.  Required  the  sura  of  an  increasing  arithmetical  pro- 
gression, having  3  for  its  first  term,  2  for  the  common  difference, 
and  the  numher  of  terms  20.     diis.  440. 

77.  Required  the  sum  of  a  decreasing  arithmetical  progres- 
sion, having  10  for  its  first  term,  -|  for  the  common  difference, 
and  the  number  of  terms  21.     Jlns.  140. 


208  Algebra. 

78.  Required  the  number  of  all  the  strokes  of  a  clock  in 
twelve  hoursy  that  is,  a  complete  revolution  of  the  index. 

Ans,  78. 

79.  The  clocks  of  Italy  go  on  to  24  hours  5  how  many  strokes 
do  they  strike  in  a  complete  revolution  of  the  index  ?    Jins,  300. 

80.  One  hundred  stones  being  placed  on  the  ground,  in  a 
straight  line,  at  the  distance  of  a  yard  from  each  other,  how  far 
will  a  person  travel  who  shall  bring  them  one  by  one  to  a 
basket,  whicJi  is  placed  one  yard  from  the  first  stone  ? 

Jins,  5  miles  and  1300  yards. 

Summation  of  Geometrical  Progressions, 

SECTION  III.     CIIAPTEB  10. 

81.  A  SERVANT  agreed  with  a  master  to  serve  him  eleven 
years  without  any  other  reward  for  his  service  than  the  pro- 
duce of  one  wheat  corn  for  the  first  year  ;  and  that  product  to 
be  sown  the  second  year,  and  so  on  from  year  to  year  till  the 
end  of  the  time,  allowing  the  increase  to  be  only  in  a  tenfold 
proportion.     What  was  the  sum  of  the  whole  produce  ? 

Jns,  111111111110  wheat  corns. 
N.  B.  It  is  further  required,  to  reduce  this  number  of  corns 
to  the  pro])er  measures  of  capacity,  and  then  by  supposing  an 
average  price  of  wheat  to  compute  the  value  of  the  corns  in 
money. 

82.  A  servant  agreed  with  a  gentleman  to  serve  him  twelve 
months,  provided  he  would  give  him  a  farthing  for  his  first 
month's  sei^ice,  a  penny  for  the  second,  and  4(1,  for  the  third, 
&c.    What  did  his  wages  amount  to  ?     ,ins.  5825/.  8s.  5|rf. 

83.  Sessa,  an  Indiaiif  having  invented  the  game  of  chess, 
shewed  it  to  his  prince,  who  was  so  delighted  with  it,  that  he 
promised  him  any  reward  he  should  ask ;  upon  wliich  Sessa 
requested  that  he  might  be  allowed  one  grain  of  wheat  for  the 
first  square  on  the  chess  board,  two  for  the  second,  and  so  on, 
doubling  continually,  to  64,  the  whole  number  of  squares ;  now 
supposing  a  pint  to  contain  7680  of  those  grains,  and  one  quarter 
to  be  worth  ll.  7s,  6d,  it  is  required  to  compute  the  value  of  the 
whole  sura  of  grains.     Jns.  /.64481488296. 


Questions  for  Practice^  209 

Simple  EqvMtioiis. 

SECTION  IV.     CHAPTER  2. 

84.  If  0?  —  4  +  6  =  8,  then  will  x  =  6. 

85.  If  4a?  —  8  =  Sx  +  20,  then  will  a>  =  28. 

86.  l{  ax  =  ah  —  a,  then  will  x  =:h  —  1. 

87.  If  2^  +  4  =  16,  then  will  x  =  6. 

88.  If  ax  +  26a  =  3c»,  then  will  x  = —  25. 

« 

89.  If  |-  =  5  +  3,  then  wil  a:  =  16. 

90.  If  ^  —  2  =  6  +4^  then  will  2x  —  6  =  18. 

o 

91.  If  a  —  —  =  c,  then  will  x  = . 

:v  a — c 

92.  If  5x  —  15  =  2x  +  6,  then  will  x  =  7. 

93.  If  40  —  6X  —  16  =  120  —  Ux,  then  will  X  =  12. 

94.  If  I  —  ^  +^  =  10,  then  willoJ  =  24. 

95.  If  "^^  +  -  =  20  — .^iril^^  then  will  x  =  23i. 

2^3  2  * 

96.  If  aJI  x  +  5  =  7,  then  will  x  =  6. 

o 

97.  If  a:  +  Ja^  +  x^  =        ^""^      ,  then  will  x  =  a  V-- 

98.  If  3aa?  +  -I  —  3  =  6a?  —  a,  then  will  x  =     ^^^"^ 


2  6a— :^6 

99.  If  x/'n+lc,  =  2  +  v^  then  will  x  =  4. 


2ffo         .,  .„  1 


100.  If  1/  +  \/«^  +  y^  = ^'^    ^,  then  will  y  = —a  V3. 

101.  If  2(|1  +  ?dL^  =  16  -  .Y±3  ^^^^  ^.^j  ^  ^  ^3^ 

102.  If  vT"  -f-  Va  +  oj  =      -^,  then  will  a:  =  — . 


103.  If  \aa  -\-  XX  z=.  ^b-^  +  a*,  then  will  x  =  ^  ^^2    . 

104.  If  1/  =  \/tt^  +  \/u2  +  x^  —  a,  then  will  a?  =  ^  —  a. 

^        ^  .  4« 

■27 


210  .9l2:ebra. 


c5' 


105.  If = -,  then  will  x  =  12. 

106.  If—,  =  —  .  then  will  x  =  S. 

107    If  — ^  =  -^  ,  then  will  ^  =  6. 
2JC+S       4ar— a 

108.  If = then  will  x  =  6. 

o  4 

109.  If  615a;  —  7x^  =  48a:,  then  will  a?  =  9. 


SECTION  IV.     CHAPTER  3, 


110.  To  find  a  number,  to  which  if  there  be  added  a  half,  a 
third,  and  a  fourth  of  itself,  the  sum  will  be  50.  ^m,  24. 

111.  A  person  being  asked  what  his  age  was,  replied  that 
1  of  his  age  multiplied  by  -^\  of  his  age  gives  a  product  equal  to 
his  age.     What  was  his  age  ?  Ms,  16. 

112.  The  sum  of  660/.  was  raised  for  a  particular  purpose  by 
four  persons.  A,  B,  C  and  D  ;  B  advanced  twice  as  murh  as  A ; 
C  as  much  as  A  and  B  togetlier  ;  and  D  as  much  as  B  and  C. 
What  did  each  contribute  ?  Ans.  60/,  120/,  ISO/,  and  300/. 

113.  To  find  that  number  whose  ^  part  exceeds  its  l  part 
by  12.  *ens,  144. 

1 14.  What  sum  of  money  is  that  whose  -|  part,  i  part,  and  ^ 
part  added  togetlier,  amounts  to  94  pounds  ?  Jns,  120/. 

115.  In  a  mixture  of  copper,  tin,  and  lead,  one  half  of  the 
wliole  —  16/6,  was  copper ;  4  of  the  whole  —  12/6.  tin  ;  and  1 
of  the  wh  de  +  4/6.  lead  :  what  quantity  of  each  was  tfieie  in  the 
composition  ?  Ans.  128/6.  of  copper,  84/6.  of  tin,  and  76/6.  of  lead. 

116.  What  number  is  that  whose  |  part  exceeds  its  |  by  72  ? 

Ms,  540. 

117.  To  find  two  numbers  in  the  proportion  of  2  to  1,  so  that 
if  4  be  added  to  each,  the  two  sums  shall  be  in  the  proportion  of 
3  to  2.  Ms.  4  and  8. 

1 18.  There  are  two  numbers  such  that  ^  of  the  greater  add^ 
to  i  of  the  less  is  13,  and  if  i  the  less  be  taken  from  ^  of  the 
greater,  the  remainder  is  nothing  ;  what  are  the  numbers  ? 

Ms.  18  and  12. 

119.  In  the  composition  of  a  certain  quantity  of  gunpowder  | 


^lestions  for  Tradict.  til 

of  the  whole  phis  10  was  nitre  ;  \  of  the  whole  minus  4|  was 
sulphur,  and  the  cfuircoal  was  |  of  the  nitre  —  2.  How  many 
pounds  of  gunpowder  were  there  ?  *^ns.  69. 

120.  A  person  has  a  lease  for  99  years ;  and  bcini^  asked 
how  much  of  it  was  already  expired,  answered,  that  two  thirds 
of  the  time  past  was  equal  to  four  fifths  of  the  time  to  come  : 
required  the  time  past.  •Ans.  54  years. 

121.  It  is  required  to  divide  the  number  48  into  two  sut  h 
parts,  that  the  one  paii;  may  be  three  times  as  much  above  20, 
as  the  other  wants  of  20.  Jins.  SZ  and  16. 

1..2.  A  person  rents  25  acres  of  land  at  7  pounds  12  shillings 
per  annum  ;  this  land  consisting  of  tw  o  sorts,  lie  rents  the  better 
sort  at  8  shillings  per  acre,  and  the  worse  at  5  :  required  the 
number  of  acres  of  tlie  better  sort.  Ans,  9. 

123.  A  certain  cistern  which  would  be  filled  in  12  minutes 
by  two  pipes  running  into  it,  would  be  filled  in  20  minutes  by 
one  alone.  Required,  in  what  time  it  would  be  filled  by  the 
other  alone.  Jins.  30  minutes. 

124.  Required  two  numbers,  whose  sum  may  be  s,  and  their 

proportion  as  a  to  ft.  dns.  — ——  and 


125.  A  privateer,  running  at  the  rate  of  10  miles  an  hour, 
discovers  a  ship  18  miles  off  making  way  at  the  rate  of  8  miles 
an  hour ;  it  is  demanded  how  many  miles  the  ship  can  run  be- 
fore she  will  be  overtaken  ?  Jns,  72. 

126.  A  gentleman  distributing  money  among  some  poor 
people,  found  he  wanted  1  Os.  to  be  able  to  give  5s.  to  each ; 
therefore  he  gives  45.  only,  and  finds  that  he  has  5s.  left : 
required  the  number  of  shillings  and  of  poor  people. 

J)is,  15  poor  people,  and  65  shillings. 

127.  There  are  two  numbers  whose  sura  is  the  6tli  part  of 
their  product,  and  the  greater  is  to  the  less  as  3  to  2.  Required 
those  numbers.  Jns.  15  and  10. 

JV*.  B.  This  question  may  be  solved  likewise  by  means  of  one 
unknown  letter. 

128.  To  find  three  numbers,  so  that  the  first,  with  half  the 
other  two,  the  second  with  one  third  of  the  other  two,  and  the 
third  with  one  fourth  of  the  other  two,  may  be  equal  to  34. 

Ms.  26,  22,  and  10. 


212  Algebra, 

129.  To  find  a  number  consisting  of  three  places,  whose 
digits  are  in  ai'ithinetical  progression  ;  if  this  number  be  divided 
by  tlie  sum  of  its  digits,  the  quotients  will  be  48  ;  and  if  from 
the  number  be  subtracted  198,  the  digits  will  be  inverted. 

Arts.  432. 

130.  To  find  three  numbers,  such  that  |  the  first,  ^  of  the  se- 
cond, and  1  of  the  third,  shall  be  equal  to  62  ;  |  of  the  first,  |  of 
the  second,  and  |  of  the  third,  equal  to  47 ;  and  \  of  the  first, 
I  of  the  second,  and  ^  of  the  third,  equal  to  38.  Jins,  24,  60,  120, 

131.  To  find  three  numbers  such  that  the  first  with  i  of  the 
sum  of  the  second  and  third  shall  be  120,  the  second  with  \  of 
the  difference  of  the  third  and  first  shall  be  70,  and  |  of  the  sum 
of  the  three  numbers  shall  be  95.  Jlns,  50,  63,  75, 

132.  What  is  that  fraction  which  will  become  equal  to -I,  if 
an  unit  be  added  to  the  numerator ;  but  on  the  contrary,  if  an 
unit  be  added  to  the  denominator,  it  will  be  equal  to  i  ?   Jlns,-^-^, 

133.  The  dimensions  of  a  certain  rectangular  floor  are  such, 
that  if  it  had  been  2  feet  broader,  and  3  feet  longer,  it  would 
have  been  64  square  feet  larger ;  but  if  it  had  been  3  feet  broader 
and  2  feet  longer,  it  would  then  have  been  68  squai'e  feet  larger : 
required  the  length  and  breadth  of  the  floor. 

Ans,  Length  14  feet,  and  breadth  10  feet. 

134.  A  person  found  that  upon  beginning  the  study  of  his 
profession  \  of  his  life  hitherto  had  passed  before  he  commenced 
his  education,  |  under  a  private  teacher,  and  the  same  time  at  a 
public  school,  and  four  years  at  the  university.    What  was  his 

?  Ans,  21  years. 

135.  To  find  a  number  such  that  whether  it  be  divided  into 
two  or  three  equal  parts  the  continued  product  of  the  parts  shall 
be  equal  to  the  same  quantity.  Ms,  6|. 

136.  There  is  a  certain  number,  consisting  of  two  digits. 
The  sum  of  these  digits  is  5,  and  if  9  be  added  to  the  number 
itself  the  digits  will  be  invei-ted.  What  is  the  number  ?    Ms,  23. 

137.  What  number  is  that  to  which  if  I  add  20  and  from  f 
this  ,sum  I  subtract  12,  the  remainder  shall  be  10  ?    Ms,  13* 


^estioiis  far  Vraciiee,  2.1^ 

Quadratic  Equations, 

SECTION  IV.     CflAPTER  5. 

138.  To  find  that  number  to  which  20  being  added,  and  from 
which  10  being  subtracted,  the  square  of  the  sum,  added  to 
twice  the  square  of  the  remainder,  shall  be  17475.       Jins.  75. 

139.  What  two  numbers  are  those,  which  are  to  one  another 
in  the  ratio  of  3  to  5,  and  whose  squares,  added  togetlier,  make 
1666  ?  Jltis,  21  and  35. 

140.  The  sum  2a,  and  the  sum  of  the  squares  2b,  of  two  num- 
bers being  given ;  to  find  the  numbers. 

^ns.  a  —  \/b  —  fl*    and  a  +  \/b  —  a*. 

141.  To  divide  the  number  100  into  two  such  parts,  that  the 
sum  of  their  square  roots  may  be  14.  •4?ts.  64  and  36. 

142.  To  find  three  such  numbers,  tliat  the  sum  of  the  fii*st 
and  second  multiplied  into  the  third,  may  be  equal  to  63 ;  and 
the  sum  of  the  second  and  third,  multiplied  into  the  first  equal 
to  28  ;  also,  that  the  sum  of  the  first  and  third,  multiplied  into 
the  second,  may  be  equal  to  55.  Jins,  2,  5,  9. 

143.  What  two  numbers  are  those,  whose  sum  is  to  the 
greater  as  11  to  7 ,  the  difference  of  their  squares  being  132  ? 

»te.  14  and  8. 


NOTES. 

There  are  many  notes  subjoined  to  the  Algebra  of  Euler  by  different  editors^  but  as  they  are  mostly 
intended  for  adepu  in  the  science,  only  a  few  of  them  are  retained  in  this  introduction. 

NOTE  1.   p.  2. 

Several  matheinatical  writers  make  a  distinction  between  Analy- 
sis and  Mgebra,  By  the  term  Analysis,  they  understand  the  method 
of  finding  those  general  rules  which  assist  the  understanding  in  all 
mathematical  investigations ;  and  by  Mgebra,  the  instrument  which, 
this  method  employs  for  accomplishing  that  end. 

2.  p.  8. 

Multiplication  is  the  taking,  or  repeating  of  one  given  number  as 
many  times,  as  the  number  by  which  it  is  to  be  multiplied  contains 
units.  Thus,  9x3  means  that  9  is  to  be  taken  3  times,  or  that  the 
measure  of  multiplication  is  3 ;  again  9  x  |  means  that  9  is  to  be 
taken  half  a  time,  or  that  tlie  measure  of  multiplication  is  |  In  mul- 
tiplication there  are  two  factors,  which  are  sometimes  called  the 
multiplicand  and  the  multiplier.  These,  it  is  evident,  may  recipro- 
cally change  places,  and  the  product  will  be  still  the  same :  for  9  x  3 
=  3x9,  and  9  x  I  —  ^  X  9.  Hence  it  appears,  that  numbers 
may  be  diminished  by  multiplication,  as  well  as  increased,  in  any 
given  ratio,  for  9  x  J  =  4|,  9  X  |^  =  1,  9  X  -^  =  f  Jo>  ^c.  The 
same  will  be  found  true  with  respect  to  algebraic  quantities;  a  x  b 
=  a6,  —  9  X  3  = —  27,  that  is,  9  negative  integers  multiplied  by  3, 
or  taken  3  times,  are  equal  to  —  27,  because  the  measure  of  multi- 
plication is  3.  In  the  same  manner,  by  inverting  the  factors,  3  posi- 
tive integers  multiplied  by  —  9,  or  taken  9  times  negatively,  must 
give  the  same  result.  Therefore  a  positive  quantity  taken  negatively, 
or  a  negative  quantity  taken  positively,  gives  a  negative  product. 

From  these  considerations,  it  will  not  be  difficult  to  shew,  that  the 
product  of  two  negative  quantities  must  be  positive.  First,  algebraic 
quantities  may  be  considered  as  a  series  of  numbers  increasing  in  any 
ratio,  on  each  side  of  nothing,  to  infinity.  Let  us  assume  a  small 
part  of  such  a  series  in  which  the  ratio  is  unity,  and  let  us  multiply 
every  term  of  it  by  —  2. 


JSlotes.  215 

5,       4,       3,        2,        I,       O,—  !,— 2,  — 3,  — 4,-^5, 

—  2,  ~  2,  —  2,  —  2,  —  2,  —  2,  —  2,  —  2,  —  2,  —  2,  —  2, 

«-  10,  -^  3,  —  6,  —  4,  —  2,  0,  +  2,  -p  4,  +  6,  +  8,  +  10, 
Here,  the  series  is  inverted,  and  the  ratio  doubled.  Further,  in  order 
to  illustrate  the  subject,  we  may  consider  the  ratio  of  a  series  effrac- 
tions between  1  and  0,  as  indefinitely  small,  till  the  last  term  being 
multiplied  by  —  2,  the  product  would  be  equal  to  0,  If,  after  this, 
the  multiplier  having  passed  the  middle  term  0,  be  multiplied  into 
any  nej^ative  term,  however  small,  between  0  and  —  1,  on  the  other 
side  of  the  series,  the  product,  it  is  evident,  must  be  positive,  other- 
wise the  series  could  not  go  on.  Hence  it  appears,  that  the  taking  of 
a  negative  quantity  negatively  destroys  the  very  property  of  negation, 
and  is  the  conversion  of  negative  into  positive  numbers.  So  that  if 
-f  X  —  =  — 5  it  necessarily  follows  that  —  X  —  gives  a  contrary 
product,  that  is,  +. 

3.  p.  10. 

All  the  prime  numbers  from  1  to  101000  are  to  be  found  in  a  Ger- 
man work,  entitled  Thoughts  on  Algebra, 

4.  p.  11. 

There  is  a  table  at  the  end  of  a  German  book  of  arithmetic,  pub- 
lished at  Leipsic  by  Poetius  in  1728,  in  which  all  the  numbers  from 
1  to  10000  are  represented  in  this  manner  by  their  simple  factors. 

5.  p.  16. 

There  are  some  numbers  with  respect  to  which  it  is  easy  to  per- 
ceive whether  they  are  divisors  of  a  given  number  or  not, 

A  given  number  is  divisible  by  2,  if  the  last  digit  is  even  ;  it  is 
divisible  by  4,  if  the  two  last  digits  are  divisible  by  4  ;  it  is  divisible 
by  8,  if  the  three  last  digits  are  divisible  by  8;  and,  in  general,  it 
is  divisible  by  2",  if  the  n  last  digits  are  divisible  by  £". 

A  number  is  divisible  by  3,  if  the  sum  of  the  digits  is  divisible  by 
3 ;  it  may  be  divided  by  6,  if,  beside  this,  the  last  digit  is  even  5  it  is 
divisible  by  9,  if  the  sum  of  the  digits  may  be  divided  by  9. 

Every  number,  that  has  the  last  digit  0  or  5,  is  divisible  by  5, 

A  number  is  divisible  by  1 1,  when  the  sum  of  the  first,  third,  fifth, 
&c.  digits  is  equal  to  the  sum  of  the  second,  fourth,  sixth,  &c.  digits. 

It  would  be  easy  to  explain  the  reason  of  these  rules,  and  to  extend 
them  to  the  products  of  the  divisors  which  we  have  just  now  con- 
sidered.   Rules  might  be  devised  likewise  for  some  other  numbers, 


-216  Mgebra, 

but  the-application  of  them  would  in  general  be  longer  than  an  actual 
trial  of  the  division. 

For  example,  I  say,  that  the  number  63704689213  is  divisible  by 
7,  because  I  find  that  the  sum  of  the  digits  of  the  number  64004245433 
is  divisible  by  7  ;  and  this  secoiid  number  is  formed,  according  to  a 
Tery  simple  rule,  from  the  remainders  found  after  dividing  by  7. 

Thus  53704689213  =  50000000000  +  3000000000  +  7u0000000 
-f  0  +  4000000  -f  600000  +  80000  -f  9000  +  200  +  10  -f  S| 
7)  50000000000 

7142857142 6 

7)  3000000000 

428571428..... 4 
7)  700000000 

0 0 

7)  00000000 

0 0 

7)  4000000 

571428 4 

7)  600000 

85714 2 

7)  80000 

•  11428 4 

7)  9000 

1285 5 

7)200 

28 4 

7)  10 

1 3 

7)3 3 

Where  the  remainders  form  the  number  64004245433,  which  is  the 
one  given  above. 

If  a,  &,  c,  d,  e,  &c.  be  the  digits  composing  any  number,  the  num- 
ber itself  may  be  expressed  universally  thus;  a  +  106  +  lO^c  + 


:N'otes.  Q\7 

lO^d  4-  10*e,  &c,  to  10.^  z;  where  it  is  easy  to  perceive  that,  if  each 
of  the  terms  a,  106,  10*  c,  &c.  be  divisible  by  w,  the  number  itself  a  + 

106  +  10* c,  &c.  will  also  be  divisible  by  n.     And,  if  — , ,  — —  > 

&c.  leave  the  remainders,  p,  q^  r,  &c.  it  is  obvious,  that  a  -f-  106  -f 
10* c,  &c.  will  be  divisible  by  w,  when  /?  +  </  +  r,  is  divisible  by  »; 
which  renders  the  principle  of  the  rule  sufficiently  clear. 

6.    p.  17. 

A  similar  table  for  all  the  divisors  of  the  natural  numbers,  from  I 
to  10000,  was  published  at  Leyden  in  1767. 

r.    p.  21. 

Though  any  definite  part  of  one  infinite  series  may  be  the  half, 
the  third,  &c.  of  a  definite  part  of  another,  yet  stiU  that  pait  bear* 
no  proportion  to  the  whole,  and  the  series  can  only  be  said,  in  that 
case,  to  go  on  to  infinity  in  a  different  ratio.  But  further,  J.  or  any 
other  numerator,  having  0  for  its  denominator,  is,  when  expanded, 
precisely  the  same  as  J. 

2             2... 
Thus,  —   =  ,,  by  division,  becomes 

2  —  2)2    (I  4-  1  4-  1,  &c.  ad  infinitum 
2  —  2 


2 
2—2 


2,  &c. 


27 


218  Jlgebra. 

8.  p.  26. 
The  rule  for  reducing  fractions  to  a  common  denominator,  may  be 
more  concisely  expressed  thus.  Multiply  the  numerator  of  each 
fraction  into  every  denominator  except  its  own,  for  a  new  numerator, 
and  all  the  denominators  to^^ether  for  a  common  denominator.  When 
this  operation  has  been  performed,  it  will  appear  that  the  numerator 
and  denominator  of  each  fraction  have  been  multiplied  by  the  same 
quantity,  and  consequently  retain  the  same  value. 

9.    p.  SI. 
Complete  tables  of  the   squares  of  natural  numbers,  from   1  to 
100000  have  been  constructed,  i.i  which  are  also  to  be  found  the  pro- 
ducts of  any  two  numbers  less  than  100000. 


ERRATA. 

Page  20,  tenth  line  from  the  top,  dele  which. 

.     39,  foui-th  line  from  the  bottom  for  y/ —  4  read  \/1I^. 
•     40,  for  nor  lead  or, 

.  169,  fifth  line  and  eighth  line  from  the  bottom  for  sou  read  sous. 
.  192,  seventeenth  line  from  the  top  for  seels  read  sells. 


